GIFT   OF 
Mrs.    W.    Barstow 


PREFACE. 


rpHE  PACKARD  COMMERCIAL  ARITHMETIC,  of  which  this  is  a 
**-  revision,  has  been  before  the  public  for  a  space  of  five  years, 
during  which  time  it  has  attained  a  large  sale  and  has  given  satis- 
faction to  its  patrons.  No  special  effort  has  been  made  to  increase 
the  sale  by  advertising  or  making  strong  statements  on  its  behalf. 
The  authors  and  publisher  have  well  understood  from  the  begin- 
ning that  the  success  of  a  text-book  to  be  used  by  intelligent 
teachers  was  not  in  any  sense  dependent  upon  promises  made  in 
advance,  or  claims  of  particular  merit  which  might  otherwise 
escape  attention.  The  book  grew  naturally  out  of  the  practice 
of  the  school-room,  and  the  manuscript  lessons  which  were  after- 
wards utilized  as  copy  were  submitted  to  the  most  exacting  tests, 
and  their  value  in  building  up  the  student  in  a  sound  knowledge 
of  principles  and  in  the  details  of  practice  thoroughly  established. 
And  the  satisfaction  expressed  by  those  who  have  used  the  book, 
has  amply  sustained  the  judgment  of  the  authors.  The  present 
revision  has  not  been  undertaken  on  account  of  any  dissatisfac- 
tion expressed  with  the  old  edition,  nor  because  it  had  not  proved 
in  the  broadest  sense  effective,  but  because  the  authors  have  felt 
that  some  additions,  and  particularly  in  the  direction  of  preparatory 
work,  would  make  it  more  acceptable  to  a  large  number  who  now 
use  it,  while  it  would  not  in  any  way  detract  from  its  utility  or 
symmetry  as  a  complete  text-book.  The  omission  of  the  funda- 
mental rules  in  the  old  edition  emphasized  the  fact  that  it  was  an 
advanced  commercial  text-book,  intended  for  schools  not  requir- 
ing primary  instruction.  It  has  come  to  our  knowledge,  however, 


IV  PREFACE. 

that  in  many  of  such  schools,  the  better  methods  of  applying  the 
fundamental  rules  would  be  acceptable,  and  that  such  an  addition 
would  make  the  book  more  useful  in  the  class-room.  In  present- 
ing this  introductory  matter,  great  care  has  been  taken  to  divest 
it  of  all  unnecessary  rules  and  exercises,  and  to  bring  it  into 
harmony  with  the  other  portions  of  the  book  for  directness  and 
forcible  application  of  principles.  The  book,  as  now  presented,  is 
deemed  to  be  a  complete  Commercial  Arithmetic,  with  little  that  is 
redundant,  and  with  all  that  is  requisite  to  establish  the  learner  in 
a  thorough  knowledge  of  commercial  usages.  A  large  number  of 
practical  examples  have  been  added  in  the  various  departments, 
not  with  a  view  of  parading  them,  but  in  order  to  satisfy  teachers 
who  wish  a  wider  field  to  cull  from.  In  making  this  addition  the 
same  rule  has  been  observed  which  did  so  much  to  commend  the 
old  edition  to  practical  teachers,  viz. :  to  admit  no  puzzles  nor 
conundrums,  and  in  every  case  to  test  the  example  by  the  busi- 
ness standard.  In  all  matters  as  to  business  customs  or  local  laws 
care  has  been  taken  to  get  information  from  the  highest  source 
and  of  the  latest  date. 

In  short,  in  presenting  this  practically  new  book  the  authors 
have  sought  to  meet  the  reasonable  expectation  of  their  friends 
and  the  public  generally  in  keeping  abreast  with  the  times,  and 
to  show  their  faith  in  honest  work. 


CONTENTS. 


PAGE 

NOTATION  AND  NUMERATION 7 

Roman  Notation . . . .  9 

ADDITION 10 

SUBTRACTION 18 

Short  method  of  finding  the  balance  of  an  account 21 

MULTIPLICATION 22 

Short  methods 20 

DIVISION 36 

Short  methods. 40 

UNITED  STATES  MONEY 41 

PROPERTIES  OF  NUMBERS 49 

Prime  Factors 51 

Common  Multiples 52 

Cancellation 54 

FRACTIONS 56 

Reduction 58 

Addition 62 

Subtraction 64 

Multiplication 65 

Division 70 

DECIMALS 77 

Reduction 81 

Addition 82 

Subtraction 84 

Multiplication 85 

Division 86 

To  find  the  value  of  goods  sold  by  the  hundred  or  thousand 88 

DENOMINATE  NUMBERS 91 

Reduction  of  Denominate  Integers 91 

Reduction  of  Denominate  Fractions 93 

Addition  of  Denominate  Numbers 97 

Subtraction  of  Denominate  Numbers 98 

Multiplication  of  Denominate  Numbers 98 

Division  of  Denominate  Numbers <• .  99 

Divisions  of  Time 100 

To  find  the  interval  of  time  between  two  dates 102 

Linear  Measures. 104 

Square  Measures. 106 

Cubic  Measure 112 

Broad  Measure 114 

Liquid  Measures 116 

Dry  Measure. 118 

Measures  of  Weight 119 

English  Money 122 

Miscellaneous  Tables 124 

Circular  Measure 125 

Longitude  and  Time 125 


VI  CONTENTS. 

PAGE 

THE  METRIC  SYSTEM 128 

Linear  Measure 128 

Square  Measure 130 

Cubic  Measure 131 

Dry  and  Liquid  Measure 132 

Weight 133 

Table  of  Equivalents 134 

Approximate  Rules 135 

ALIQUOT  PARTS 139 

PERCENTAGE 141 

PROFIT  AND  Loss 150 

DISCOUNTS 153 

BILLS 157 

COMMISSION  AND  BROKERAGE 166 

INTEREST 171 

Accurate  Interest 184 

PROBLEMS  IN  INTEREST 186 

PRESENT  WORTH  AND  TRUE  DISCOUNT 190 

COMPOUND  INTEREST 193 

COMMERCIAL  PAPER 197 

BANK  DISCOUNT 203 

PARTIAL  PAYMENTS 207 

United  States  Rule 207 

Mercantile  Rules 210 

RATIO  AND  PROPORTION 213 

Simple  Proportion 214 

Compound  Proportion 217 

INSURANCE 219 

Fire  Insurance 220 

Marine  Insurance 221 

EXCHANGE 225 

Domestic  Exchange 226 

Foreign  Exchange 229 

EQUATION  OF  ACCOUNTS 237 

When  the  items  are  all  debits  or  all  credits 237 

When  the  account  contains  both  debit  and  credit  items 246 

Equation  of  Account  Sales 251 

ACCOUNTS  CURRENT 255 

STOCKS  AND  BONDS 263 

New  York  Stock  Exchange 266 

TAXES 276 

DUTIES 279 

PARTNERSHIP 286 

NATIONAL  BANKS 300 

SAVINGS  BANKS 303 

LIFE  INSURANCE 306 

APPENDIX 317 

Greatest  Common  Divisor 317 

Annual  Interest 319 

New  Hampshire  Rule 321 

Vermont  Rule 322 

Storage 323 

Alligation 327 

Square  Root 331 

Cube  Root 332 

Mensuration 334 

General  Average 337 

Foreign  Weights  and  Measures 342 

Detection  of  Errors  in  Trial  Balances. .  .  344 


ARITHMETIC. 


NOTATION    AND    NUMERATION. 

1.  Arithmetic  is  the  science  of  numbers  and  the  art  of  com- 
putation by  them. 

2.  A  Unit,  or  Unity,  is  one,  or  a  single  thing  ;  as  one,  one 
foot,  one  dollar. 

3.  A  Number  is  a  unitv  or  a  collection  of  units  ;  as  one, 
four,  three  feet,  five  dollars.     Numbers  are  expressed  by  words, 
by  letters,  and  by  figures. 

4.  Notation  is  a  system  of  representing  numbers  by  symbols. 
There  are  two  methods  of  notation  in  use,  the  Roman  and  the 
Arabic. 

5.  Numeration  is  a  system  of  naming  or  reading  numbers. 

6.  The  Arabic  method  of  notation  employs  ten  characters  or 
figures,  viz.  : 


One,         Two,      Three,      Four,       Five,        Six,       Seven,     Eight,      Nine,       Zero, 

The  first  nine  of  the  above  are  called  significant  figures,  because  each, 
standing  by  itself,  represents  a  value,  or  denotes  some  number.  They  are 
also  called  digits,  from  the  Latin  word  digitus,  which  means  a  finger. 

The  last  one  is  called  zero,  naught,  or  cipher,  because  when  standing 
alone  it  has  no  value,  or  signifies  nothing, 


NOTATION     AXD      JV  UM  ERATION.  [Art.  6. 

FRENCH  AND  AMERICAN  NUMERATION  TABLE. 


s    £         3    « 
-1    .2  «     § 

5      '  '•-'S3 

ac      CH  DO      (S 

ulUil! 

H     ^    W     En     ^     K     H 

4  S,  1   9  5,  7  3  2,  4  3  6,  8  0  7,  5  9  3. 

••  7Yh  period'    eth  period,  Sth  period,  4th  period,  3d  period,    2d  period,    1st  period. 
Quintillions.  Quadrillions.  Trillions.     Billions.       Millions.    Thousands.      Units. 

ENGLISH  NUMERATION  TABLE. 


H  £ 

508642195732436807593 

Billions.  Millions.  Units. 

NOTE. — It  will  be  observed  by  a  comparison  of  the  French  and  English 
systems,  that  numbers  consisting  of  nine  figures  or  less  are  read  the  same. 

7.  Copy  and  read  the  following  numbers  : 

73  102  616  1064  8174  12741 

69  333  348  3604  8006  20809 

48  570  222  4364  7070  47038 

90  895  843  7208  3300  68605 

8.  Express  by  figures  the  following  : 


1.  Nineteen. 

2.  Twenty-two. 
S.  Forty-six. 

4.  Sixty-eight. 

5.  Ninety-two. 

6.  Eighty-seven. 


7.  One  hundred  forty-four. 

8.  Three  thousand  sixteen. 

9.  Four  thousand  forty-four. 

10.  Six  million  two  thousand  six. 

11.  Sixteen  million  eight  hundred  two. 

12.  Eighty-seven  thousand  sixty-two. 


Art.  9.] 


ROMAN    NOTATION. 


ROMAN     NOTATION. 

9.  In  the  Roman  Notation,  seven  capital  letters  are  used 
to  express  numbers,  as  follows  : 

I         V        X       L  C  D  M 

One,          Mve,  Ten,         fifty,        One  Hundred,  Five  Hundred,  One  Thousand. 

Other  numbers  are  expressed  by  combining  the  letters  according 
to  the  following  principles  : 

1.  If  a  letter  is   repeated,  its  value  is   repeated.     Thus,  III  represents 
three  ;  XX,  twenty  ;  CCC,  three  hundred. 

2.  If  a  letter  of  less  value  is  placed  before  one  of  greater  value,  the  less  is 
taken  from  the  greater.     Thus,  IV  represents  four  ;  IX,  nine  ;  XL,  forty. 

3.  If  a  letter  of  less  value  is  placed  after  one  of  greater  value,  the  less  is 
added  to  the  greater.     Thus,  VI  represents  six  ;  XI,  eleven  ;  LX,  sixty. 

4.  A  bar  (~~)  placed  over  a  letter  increases  its  value  a  thousand  times. 
Thus,  X  represents  ten  thousand  ;  M,  one  million. 

The  Roman  Notation  is  used  for  numbering  dials,  chapters,  pages,  etc. 


10. 


TABLE  OF  ROMAN  NOTATION. 


Roman. 

Arabic. 

Eoman. 

Arabic. 

Roman. 

Arabic. 

Roman. 

Arabic. 

I, 

I. 

IX, 

9. 

XX, 

20. 

xc, 

90. 

II, 

2. 

x, 

10. 

XXI, 

21. 

c, 

100. 

III, 

3. 

XIII, 

13. 

XXX, 

30. 

CCC, 

300. 

IV, 

4. 

XIV, 

14. 

XL, 

40. 

D, 

500. 

v, 

5. 

XV, 

15. 

L, 

50. 

DCC, 

700. 

VI, 

6. 

XVIII, 

18. 

LX, 

60. 

M, 

1000. 

VIII, 

8. 

XIX, 

19. 

LXXX 

,  80. 

MD, 

1500. 

11.  Express  by  Roman  notation  : 


1.  Eighteen. 
2.   Thirty-six. 
3.  Forty-eight. 
4.  Seventy-six. 
5.   Sixty-four. 

6.  Eighty-seven. 
7.  Three  hundred  sixty. 
8.  Six  hundred  forty-nine. 
9.  Five  hundred  eighty-eight. 
10.  Two  thousand  sixty-two. 

12.  Express  by  Arabic  notation  : 

1.  LXXVII. 

6.  DCCLXVI. 

11.  MMCC 

2.  CCXIX. 

7.  DCXLIV. 

12.  MMDC 

3.  XCVIII. 

8.  DCXLIV. 

IS.  MMCC 

4.  CCCLIV. 
5.  DCXXVI. 

9.  MDCXLVI. 
10.  MCCLXXIX. 

14.  MMDC 
15.  MDCCC 

11. 

12. 

IS. 
14. 
15. 


584. 

777. 

1638. 

1886. 

80000. 


ADDITION. 


13.  The  Sum  or  Amount  of  two  or  more  numbers  is  a 
number  which  contains  as  many  units  as  all  the  numbers  com- 
bined. 

14.  Addition  is  the  process  of  finding  the  sum  of  two  or 
more  numbers. 

15.  The  sign  of  addition  is  -f ,  and  is  read  plus. 

16.  The  sign  of  equality  is   =,   and   is   read   equals,   or   is 
equal  to. 

Thus,  6  +  2  =  8  is  read  6  plus  2  equals  8,  or  the  sum  of  6  and  2  is 
equal  to  8. 

17.  The  sign  of  dollars  is  $ ;  of  cents  ^,  ct.,  or  cts. 

18.  To  find  the  sum  of  two  or  more  numbers. 

Ex.    Find  the  sum  of  416,  578,  695. 

OPERATION.  ANALYSIS.— Write  the  numbers  so  that  like  units  stand 

416  in  the  same  column  and  begin  to  add  at  the  right.     The 

578  sum  of  the  units  (6  +  8  +  5)  is  (14,  19)  19  units,  equal  to 

ggg  1  ten  9  units.     Write  the  9  units  under  the  column  of  units, 

and  add  the  1  ten  to  the  column  of  tens,  obtaining  for  the 

1689  Sum.       sum  (2,  9,  18)  18  tens,  equal  to  1  hundred  8  tens.    Write 

the  8  tens  under  the  column  of  tens,  and  add  the  1  hundred 

to  the  column  of  hundreds,  obtaining  for  the  sum  (5,  10,  16)  16  hundreds, 

equal  to  1  thousand  6  hundreds,  which  write  in  the  hundreds'  and  thousands' 

places.     Hence,  the  entire  sum  is  1689. 

NOTES. — 1.  Write  the  numbers  in  vertical  lines.      Irregularity   in  the 
placing  of  figures  is  the  cause  of  many  errors. 

2.  Think  of  results  and  not  of  the  numbers  themselves.     Thus  in  the 
above  example,  do  not  say  6  and  8  are  14  and  5  are  19,  but  14,  19. 

3.  To  avoid  repeating  the  work,  in  case  of  interruption,  write  the  figures 
to  be  carried  in  pencil  underneath. 


Art.  19.]  ADDITION.  11 

19.  RULE. —  Write  the  numbers  to  be  added  so  that  like 
units  stand  in  the  same  column. 

Commencing  at  the  right,  add  each  column  separately, 
and  if  the  sum  is  less  than  10,  write  it  under  the  column 
added. 

If  the  sum  of  any  column  is  10  or  more  than  10,  write 
the  right-hand  figure  under  the  column  added,  and  add 
the  remaining  figure  or  figures  to  the  next  column. 

PROOF. — Find  the  sum  by  adding  the  columns  in  the 
opposite  direction,  thus  forming  new  combinations  of  fig- 
ures. If  the  results  agree,  th  e  work  is  probably  correct. 

EXAMPLES. 

20.  Copy  or  write  from  dictation  and  add  the  following  : 

(1}  (2}  (3)  (4)  (6)  (6) 

789  682  1234  1357  7812  9876 

123  109  5678  9135  3625  6789 

456  375  9012  8642  •  .4875  9787 

246  488  3456  4109  '9850  8678 

(7)  (8)      (9)  (10)  (11)  (12) 

568  431  9672  7812  8796  808 

134  866  8738  1357  809  7612 

680  219  4126  404  4205  37 

419  581  1886  9686  6666  4123 

723  49  7143  8072  7777  2264 

842  376  8275  9706  8088  7714 

906  408  9325  5555  4144  9008 

294  792  4444  2009  9995  3348 

21.  There  is  nothing  of  more  importance  to  the  student  than 
the  ability  to  add  a  column  of  figures  easily,   accurately,  and 
rapidly.      In  order  that  his  labor  may  be  lightened  and  much 
valuable  time  saved,  not  only  in  after-life  but  in  his  school  work, 
he  should  have  various  kinds  of  daily  drill  exercises  in  addition, 
especially  in  the  earlier  part  of  his  course  of  study.     The  follow- 
ing suggestions  will  be  found  valuable  in  securing  accuracy  and 
rapidity : 


12  ADDITION.  [Art.  22. 

22.  The  45  simple  combinations  should  be  used  as  an  exer- 
cise in  addition.  They  may  be  copied  on  the  blackboard  in  the 
following  or  in  irregular  order,  and  the  sum  should  be  announced 
by  the  student  at  sight : 

123243543654765 
121212123123123 

48765987659876!* 
412341234523453 


8 
4 

M 
I 

5 

6 
6 

9 
4 

8 
5 

7 
6 

9 
5 

8 
6 

7 
7 

9 
6 

8 

7 

9 

7 

8 
8 

9 

8 

9 
9 

23.  The  above  should  be  supplemented  by  exercises  similar 
to  the  following  : 

74         64         44         94         34         24         14         54         84 


45         75         35         15         55         65         95         25         85 
_9_9_9_9_9_9_9  9  9 

It  is  just  as  easy  to  add  74  and  8  as  4  and  8.  It  should  be  impressed  on 
the  mind  of  the  student  that  4  and  8  when  added  always  produce  2  in  units' 
place,  whatever  the  number  of  tens,  and  the  tens  are  increased  by  1.  If  the 
student  is  thoroughly  drilled,  he  will  not  hesitate  when  near  the  end  of  the 
column  or  when  the  sum  is  above  20  or  30. 

24.  Make  combinations  of  10,  20,  30,  or  other  numbers  as 
often  as  possible,  and  add  them  as  single  numbers. 

Thus,  add  9  and  1,  8  and  2,  5  and  5,  4,  3,  and  3,  etc.,  as  10;  7  and  2,  6, 
2,  and  1,  etc.,  as  9;  2  and  3,  4  and  1,  2,  2,  and  1,  as  5;  8,  7,  and  5,  9,  7,  and 
4,  etc.,  as  20;  etc.,  etc. 

In  Example  1,  Art.  27,  think  only  of  the  following  results:  9,  19,  30,  50. 

Drill  on  the  following  and  similar  combinations  (10,  20,  30, 
etc.  )  until  the  student  can  announce  the  sums  at  sight  : 

321453822612213 
434111176281433 
355546112217464 


Irt. 

24.] 

ADDITION.                                                    K 

7 

6 

6 

8 

4 

8 

9 

4 

3 

2 

5 

8 

9 

7 

9 

6 

8 

5 

4 

7 

3 

2 

i% 

8 

9 

7 

7 

8 

7 

7 

7 

6 

9 

8 

9 

9 

9 

9 

9 

9  ' 

8 

5 

3 

6 

4 

1 

1 

1 

3 

3 

6 

8 

4 

7 

7 

7 

6 

9 

8 

9 

2 

4 

3 

3 

3 

6 

4 

4 

7 

4 

7 

M 
f 

6 

8 

4 

3 

2 

2 

1 

3 

6 

1 

4 

4 

4 

7 

8 

7 

9 

8 

4 

3 

4 

3 

1 

2 

7 

8 

2 

5 

9 

9 

8 

5 

9 

When  a  figure  or  number  comes  between  two  numbers  that  make  10,  think 
of  the  total  at  once;  as  7  +  G  +  3  =  16,  6  +  7  +  4  =  17. 

NOTES. — 1.  When  three  figures  are  in  regular  order,  the  sum  may  be  found 
by  multiplying  the  middle  figure  by  3.  Thus,  9  +  8  +  7  -  24  (3  x  8) ; 
5  +  6  +  7=  18;  6  +  7  +  8  =  21. 

2.  When  five  figures  are  in  regular  order,   the  sum  may  be  found  by 
multiplying  the  middle  figure  by  5.     Thus,  1  +  2  +  3  +  4  +  5  =  15  (ox  3); 
5  +  6  +  7  +  8  +  9  =  35. 

3.  When  a  figure  is  repeated  several  times,  multiply  it,  instead  of  adding. 

4.  Do  not  think  of  numbers  between  10  and  20  as  a  certain  number  of 
units  and  1  ten  as  they  are  named,  but  as  1  ten  and  a  certain  number  of  units. 
Thus,  think  of  14  as  1  ten  and  4  units  (onety-four,  or  one-four),  not  4  units 
and  1  ten  (fourteen). 

5.  Add  downwards,  for  then  the  sum  is  found  just  where  it  should  be 
placed — at  the  foot  of  the  column.     In  proving  results,  add  upwards. 

25.  Adding  two  columns  at  once. — Drill  on  the  follow- 
ing or  similar  combinations  of  numbers  of  two  figures  each,  until 
the  student  can  announce  the  sums  at  sight : 

12         24        45         24        37         41         37         62        27        57 
16         33         33         26         42         58         45         34        48         27 

When  the  above  have  been  mastered,  give  exercises  containing 
three  or  more  numbers,  as 

:  12  16  22  24  19  42  56  37  51  27 
31  29  33  36  31  24  21  33  25  34 
42  38  56  22  44  17  24  28  38  45 

26.  The  following  " magic  square"  may  be  used  as  a  drill 
exercise  in  addition.     The  sum  downwards,  from  left  to  right,  or 
diagonally  is  54351.     To  vary  the  exercise,  the  teacher  may  die- 


14 


ADDITION. 


[Art,  26. 


tate,  to  be  added,  all  the  numbers  but  one  in  any  line  or  column. 
The  sum  can  be  found  by  subtracting  the  number  omitted  from 
54351.  The  sum  of  the  digits  of  any  result  will  be  a  multiple 
of  9. 


4536 

9477 

3726 

8667 

2916 

7857 

2106 

7047 

1296 

6237 

486 

567 

4617 

9558 

3807 

8748 

2997 

7938 

2187 

7128 

1377 

5427 

5508 

648 

4698 

9639 

3888 

8829 

3078 

8019 

2268 

6318 

1458 

1539 

5589 

729 

4779 

9720 

3969 

8910 

3159 

7209 

2349 

6399 

6480 

1620 

5670 

810 

4860 

9801 

4050 

8100 

3240 

7290 

2430 

2511 

6561 

1701 

5751 

891 

4941 

8991 

4131 

8181 

3321 

7371 

7452 

2592 

6642 

1782 

5832 

81 

5022 

9072 

4212 

8262 

3402 

3483 

7533 

2673 

6723 

972 

5913 

162 

5103 

9153 

4293 

8343 

8424 

3564 

7614 

1863 

6804 

1053 

5994 

243 

5184 

9234 

4374 

4455 

8505 

2754 

7695 

1944 

6885 

1134 

6075 

324 

5265 

9315 

9396 

3645 

8586 

2835 

7776 

2025 

6966 

1215 

6156 

405 

5346 

EXAMPLES. 


27.  Copy  or  write  from  dictation  and  add  the  following  : 


w 

4) 
5  f 

15 

w 

13 

(3) 
12  85] 
21  96  1 

(4) 

123 

456 

382 

648 

3  ) 

(  95  ) 

(44 

21 

789 

584 

7'f  1 

30 

(62) 

20 

3M66 

462 

765 

8  ) 
3  f 

40 

36 

•j2 

l« 

315 
829 

406 
483 

6) 

eUo 

55. 

r  68) 

54) 

•{? 

1- 

918 
234 

163 

852 

1 

j   y 

8) 

(46) 

63  77 

49 

789 

574 

w 

(7) 

W 

m 

(«j 

(") 

(12) 

(18) 

48 

71 

39 

12 

77 

312 

514 

376 

13 

43 

34 

34 

88 

123 

627 

499 

82 

36 

46 

56 

66 

456 

842 

678 

67 

94 

25 

78 

99 

789 

462 

437 

54V  N 

dfe    » 

fc  83 

f90% 

,  41. 

987 

460 

245 

87 

25 

31 

89 

63 

654 

329 

536 

43 

38 

63 

76 

74 

321 

411 

984 

Art.  27.]  ADDITION.  15 

(14)  (15)  (16)  (17)  (18)  (19) 

1234  4121  1728  3416  17642  18114 

5678  1865  5280  4725  176  285 

9212  3760  2246  8850  20048  28510 

3456  4825  4153  4975  248  30048 

9753  7145  4839  2137  24800  400 

8642  3333  2437  8910  1149  17512 

7531  7163  4627  2048  1216  8 

1594  4943  7342  175  385  14150 

7777  7289  8916  1075  19175  30032 

20.  Find   the   sum   of    the   following    numbers :     Forty-five 
thousand  forty-five  ;  sixteen  thousand  three  hundred  sixty ;   one 
hundred   sixty-seven   thousand ;    eight   hundred   fifty   thousand 
ninety-two  ;  nine  million  twenty-four. 

21.  46  +  72  +  89  +  93  +  75  +  31  +  58  -f  45  +  52  =  ? 
82.  376  +  416  -f  287  +  123  +  456  +  789  +  916  +  328  =  ? 

23.  42  +  175  +  287  +  56  +  63  +  324  +  189  +  172  +  96  =  ? 

24.  365  +  1728  -f  64  +  172  +  89  +  38  +  9  +  5280  +176  =  ? 

25.  A  bushel  of  corn  weighs  56  pounds,  a  bushel  of  rye  56 
pounds,  a  bushel  of  wheat  60  pounds,  a  bushel  of  barley  45  pounds, 
a  bushel  of  oats  32  pounds,  and  a  bushel  of  buckwheat  48  pounds. 
What  would  be  the  total  weight  of  one  bushel  of  each  of  the  above 
grains  ? 

26.  A  farmer  raises  375  bushels  corn,  419  bushels  barley,  849 
bushels  wheat,   668  bushels  oats,   957   bushels  barley,  and  389 
bushels  rye.     Find  how  many  bushels  in  all. 

27.  Find  the  total  distance  around  a  rectangular  field  1728 
feet  long  and  1683  feet  wide. 

28.  An  exporter  of  provisions  buys  187  barrels  hams,  428  bar- 
rels shoulders,  475  barrels  pork,  229  barrels  beef,  and  392  barrels 
bacon.     How  many  barrels  in  all  ? 

29.  In  an  orchard  there  are  375  apple  trees,  416  pear  trees, 
37  quince  trees,  98  cherry  trees,  238  peach  trees,  and  276  plum 
trees.     How  many  trees  in  all  ? 

SO.  A  man  pays  for  a  house  and  lot  $6375.  For  repairs  as 
follows :  mason-work,  $68 ;  plumbing,  $78 ;  carpenter-work, 
$164 ;  painting  and  decorating,  $277.  For  how  much  must  he 
sell  it  to  gain  $567  on  the  total  cost  ? 


16  ADDITION.  [Art.  27. 

31.  A  manufacturer  sells  on  Monday  2387  barrels  flour,  on 
Tuesday  2618  bbls.,  on  Wednesday  2178  bbls.,  on  Thursday  2125 
bbls.,  on  Friday  2348  bbls.,  and  on  Saturday  2496  bbls.     How 
many  does  he  sell  during  the  week  ? 

32.  Find  the  total  number  of  pounds  of  tobacco  produced  in 
the  following  states  in  1879  :  Kentucky,  171,121,134;  Virginia, 
80,099,838;  Pennsylvania,  36,957,772;    Ohio,  34,725,405;  Ten- 
nessee,   29,365,052;     North    Carolina,    26,986,448;     Maryland, 
26,082,147;     Connecticut,    14,044,652;     Missouri,    11,994,077; 
Wisconsin,  10,878,463. 

Add  the  following  numbers  as  they  stand,  from  left  to  right, 
and  from  right  to  left.  [In  making  out  bills  and  in  other  com- 
mercial operations,  a  great  deal  of  time  can  be  saved  by  adding  in 
this  manner,  without  re-arranging  the  numbers.] 

S3.  17,  27,  36,  14,  43,  42,  65,  73,  81,  35. 

84.  176,  340,  203,  62,  177,  96,  398,  75,  148,  96. 

35.  137,  414,  528,  345,  678,  975,  864,  357,  121,  234. 

S6.  6716,  512,  375,  475,  3842,  5927,  3875,  17525. 

37.  2345,  16,  375,  4218,  376,  7,  8475,  247,  39. 

38.  123427,  34825,  775,  716,  8976,  37412,  567356,  39723. 

NOTE. — In  tally-sheets  of  pounds,  gallons,  yards,  feet,  etc.,  for  conveni- 
ence in  adding,  place  10  (20  or  30)  numbers  in  each  column  as  in  the  following 
example.  (See  Note  3,  Art.  .24.)  Add  the  totals  from  left  to  right. 

39.  Find  the  total  weight  of  the  following  100  boxes  of  cheese: 


67 

64 

62 

61 

06 

68 

64 

62 

61 

60 

61 

67 

60 

64 

/»/*j 

o< 

60 

63 

61 

68 

64 

60 

63 

63 

64 

66 

65 

67 

61 

60 

63 

63 

60 

62 

61 

68 

64 

65 

66 

63 

67 

62 

62 

61 

65 

61 

66 

63 

67 

62 

65 

64 

61 

60 

68 

66 

64 

63 

67 

69 

66 

66 

65 

62 

61 

68 

67 

64 

63 

66 

66 

65 

66 

65 

64 

67 

67 

68 

66 

63 

61 

61 

68 

64 

62 

65 

60 

61 

60 

62 

65 

62         64        66       _66         63      _61         62      __67      _68      _69 
***       ***       ***       ***       ***       ***       #**       ***       #**       *** 

**** 


Art.  27.] 


ADDITION. 


17 


40.  Find  the  total  estimated  value  of  the  following  crops  for 
the  year  1884:    Corn,  $640,735,589;  wheat,  $330,861,254;  rye, 
$14,855,255;  oats,  $161,528,470;  barley,  $29,781,155. 

41.  Find  the  total  number  of  bales  of  cotton  produced  in  the 
following  states  in  1879  :    Mississippi,  955,808  ;  Georgia,  814,441 ; 
Texas,   803,642;    Alabama,   699,654;    Arkansas,  608,256  ;  South 
Carolina,  522,548;  Louisiana,  508,569;  North  Carolina,  389,598. 

J$.  Find  the  total  number  of  bushels  of  wheat  produced  in 
the  following  states  in  1879  :  Illinois,  51,136,455 ;  Indiana, 
47,288,989;  Ohio,  46,014,869;  Michigan,  35,537,097;  Minne- 
sota, 34,625,657;  Iowa,  31,177,225;  California,  28,787,132; 
Missouri,  24,971,727;  Wisconsin,  24,884,689. 

Complete  the  following  statements  by  adding  downwards  and 
from  left  to  right.  The  sums  of  the  totals  should  be  equal. 

43.  SALES  FOR  THE  WEEK  EN-DING  JAN.  21,  1887. 


Days. 

Goods. 

Shoes. 

Furniture. 

Books. 

Crockery. 

Millinery. 

Totals. 

Monday 

64718 

17642 

37640 

117  13 

92  17 

Ill  40 

****  ** 

Tuesday  

35625 

97.20 

184.00 

6472 

3815 

5765 

***  ** 

Wednesday  
Thursday  
Friday 

716.40 
828.30 
316  17 

200.48 
227.59 

8714 

417.65 

476.00 
148  12 

156.25 
171.14 
10080 

127.16 

287.80 

7744 

328.40 
116.38 
7940 

****^** 
****^** 
***  ** 

Saturday  

929.12 

249.67 

417.13 

426.00 

225.98 

429.75 

****  ** 

Totals 

****  ** 

****  ** 

****  ** 

****  ** 

***  ** 

****  ** 

****  ** 

' 

44.  SALES  FOR  1887. 


Months. 

Domestics. 

White 
Goods. 

Notions. 

Woolens. 

Dress 
Goods. 

Totals. 

January    

12248  10 

6478  10 

3429.12 

811740 

5276  40 

*****  ** 

February  

13375  16 

7149  37 

4176  19 

9190  56 

6189  24 

*****  ** 

March 

17177  48 

8214  92 

4375  94 

8237  41 

3416  48 

*****  ** 

April 

15119  43 

7175  12 

4040  40 

7116  40 

5255  90 

*****  *# 

May 

16284  19 

813437 

4287  38 

3697  82 

1716  32 

*****  ** 

June 

13484  25 

6375  28 

3343  72 

2419  38 

2100  58 

*****  ** 

July  

9119  47 

4569  34 

2417  75 

3129  16 

3269  43 

*****  ** 

August  

10219  36 

5162  12 

3412  96 

4289  50 

3716  96 

*****  ** 

September  .    . 

14196  42 

70QQ  4ft 

4.198  4Q 

5391  42 

4128  31 

*****  ** 

October  
November  

13184.16 
14174  12 

6517.58 

7288  49 

3692,29 
4140  50 

7214.97 
8345  68 

6247.39 
711640 

*****  ** 
*****  ** 

December  

13184.9G 

6697.13 

3990.82 

8175.75 

7348.19 

*****^** 

Totals  

******  ** 

*****  ** 

*****  ** 

*****  ** 

*****  ** 

******  ** 

SUBTRACTION. 


2S.  The  difference  between  two  numbers  is  a  number  which,, 
added  to  the  smaller,  will  produce  a  result  equal  to  the  greater. 

29.  Subtraction  is  the  process   of  finding  the   difference 
between  two  numbers. 

The  greater  of  two  numbers  whose  difference  is  required  is  called  the 
minuend,  and  the  smaller  the  subtrahend.  The  result  is  called  the  re- 
mainder. 

30.  The  sign  of  subtraction  is  — ,  and  is  read  minus  or  less. 

Thus,  8  —  5  is  read  8  minus  5,  or  8  less  5,  and  means  that  5  is  to  be  taken 
from  8. 

31.  To  find  the  difference  between  two  numbers. 
Ex.  Find  the  difference  between  967  and  384. 

OPERATION.  ANALYSIS. — Write    the    smaller  number  under  the 

967  Minuend.  greater  so  that  units  are  under  units,  tens  under  tens, 
384  Subtrahend.  etc'  Commence  to  subtract  at  the  right.  4  units  from 
~T7  7  units  are  3  units,  which  write  below  the  line  under  the 

58d   Kemamder.       coiumn  of  units      gince  3  tens  cannot  be  taken  from  6 
tens,  take  1  hundred  from  9  hundreds,  leaving  8  hun- 
dreds, and  add  it  (1  hundred  =  10  tens)  to  the  6  tens,  making  16  tens.    8  tens 
from  16  tens  are  8  tens,  which  write  under  the  column  of  tens.     3  hundreds 
from  8  (9  —  1)  hundreds  arc  5  hundreds.     Hence  the  result  is  583. 

Instead  of  subtracting  1  from  the  figure  of  the  upper  number  of  the  next 
higher  order  when  it  has  been  necessary  to  add  10  to  the  figure  of  the  minu- 
end, some  persons  add  1  to  the  figure  of  the  lower  number  of  the  next  higher 
order.  This  method  depends  on  the  principle  that  adding  equivalent  num- 
bers to  both  minuend  and  subtrahend  does  not  affect  the  remainder. 

In  practice,  do  not  think  of  explanations,  nor  say  4  from  7  is  3,  etc.,  but 
think  only  of  results  and  write  them  at  once.  Thus,  in  the  above  example, 
say  or  think  only  3,  8,  5. 


Art.  32.]  SUBTRACTION.  19 

32.  RULE. —  Write  the  subtrahend   under  the  minuend 
so  that  units  of  the  same  order  stand  in  the  same  cuiumn. 

Commencing  at  the  right,  subtract  each  figure  in  the 
lower  number  from  the  one  above  it,  and  write  the  differ- 
ence in  the  line  below. 

If  any  figure  is  greater  than  the  one  above  it,  add  10  to 
the  latter,  perform  the  subtraction,  and  then  consider  the 
next  figure  in  the  upper  number  decreased  by  1  (or,  con- 
sider  the  next  figure  in  the  lower  number  increased  by  1). 

EXAMPLES. 

33.  Find  the  difference  between 

1.  8716  and  4379.  11.  80706040  and  23456789. 

2.  917642  and  9819.  12.  76483672  and  87132191. 

3.  64321  and  23456.  18.  123456789  and  9897960. 

4.  428165  and  317618.  14.  72081099  and  87643229. 

5.  9371641  and  876543.  15.  16417528  and  90716801. 

6.  7642878  and  6789119.  16.  43184296  and  37529510. 

7.  8090403  and  7090508.  17.  100010001  and  9890978. 

8.  6380912  and  5270937.  18.  30040050  and  29917168. 

9.  7654321  and  1234567.  19.  20103040  and  19181746. 
10.  7060509  and  6987969.  20.  40020003  and  20807064. 

Find  the  difference  between  the  numbers  in  each  of  the  fol- 
lowing groups.  [In  all  of  these  cases  the  subtrahend  is  placed 
above  the  minuend,  the  purpose  being  to  give  the  student  practice 
in  subtracting  doivnward  rather  than  upward,  as  the  general  cus- 
tom is.  It  is  often  requisite  in  business  to  perform  the  work  in 
this  way,  and  the  accountant  should  practice  both  methods.] 


(21)  (22)  (23)  (24)  (25) 

76534  19827  26347  72016  12345  81907 

81279  84362  71356  99385  54321  94371 

(27)  (28)  (29)  (30)  (31)  (32) 

12467  31617  46789  24681  46897  36478 

75112  42131  50000  30502  50901  41516 


20 


SUBTRACTION. 


[Art.  33. 


83.  There  were  50017  post-offices  in  the  United  States  in  1884 
and  51252  in  1885.  What  was  the  increase  during  the  year  ? 

34.  In    1880,    the    population    of    the    United    States    was 
50,152,866,    and   in    1870,  38,558,371.     What   was   the  increase 
during  the  decade  ? 

35.  The   area   of   Alaska   is   369,529,600  acres.     How   much 
greater  is  it  than  Texas,  whose  area  is  175,587,840  acres  ? 

86.  The  public  debt  of  the  United  States  Nov.  1,  1885,  was 
$1,447,657,568,  and  Nov.  1, 1886,  $1,354,347,947.  What  was  the 
reduction  of  the  debt  during  the  year  ? 

37.  The  gross  weights  (weights  of  barrels  and  sugar)  and  tares 
(weights  of  barrels)  of  ten  barrels  of  sugar  are  as  follows:  326-19, 
332-19,  307-18,  321-18,  324-19,  330-19,  313-18,  313-19,  317-17, 
327-19.  Find  the  total  net  weight. 

NOTE. — Find  the  total  gross  weight  and  total  tare,  and  then  the  difference, 
or  the  total  net  weight. 

Population  of  the  following  cities  of  the  United  States  in  1880: 


New  York, 
Philadelphia,  - 
Brooklyn, 
Chicago, 
Boston,  - 
St.  Louis, 
Baltimore, 
Cincinnati, 
San  Francisco, 
New  Orleans,  - 
Cleveland, 
Pittsburg, 


1,206,590 
846,984 
580,370 
503,304 
362,535 
350,522 
332,190 
255,708 
233,956 
216,140 
160,142 
156,381 


Buffalo,  - 
Washington,  - 
Newark, 
Louisville, 
Jersey  City,    - 
Detroit, 
Milwaukee,     - 
Providence,     - 
Albany, 
Rochester, 
Allegheny, 
Indianapolis,  - 


155,137 
147,307 
136,400 
123,645 

120,728 

116,342 

115,578 

104,850 

90,903 

89,363 

78,681 

75,074 


88.  What  is  the  total  population  of  the  first  column  of  the 
above  cities  ?  Of  the  second  column  ?  What  is  the  total  popula- 
tion of  all  ? 

39.  What  is  the  difference  between  the  sums  of  the  first  and 
second  columns  ? 

40.  How  much  does  the  total  population  of  New  York,  Brook- 
lyn, Newark,  Jersey  City,  Hoboken  (30,999),  Yonkers  (18,892), 
and  Long  Island  City  (17,117)  lack  of  being  2,500,000  ? 

4J.  How  much  does  the  total  population  of  Pittsburg  and 
Allegheny  exceed  that  of  San  Francisco  ? 


Art.  34.J 


S  UB  TR  A  C  TIO  N . 


21 


34.    Short    method    of    finding    the    balance    of    an 
account. 

Ex.  Find  the  balance  of  the  following  ledger  account : 
Dr.  C.  E.  &  W.  F.  PECK.  Or. 


1882. 

1882. 

Mar. 

16 

Merchandise. 

1192 

97 

Apr. 

22 

Cash. 

800 

M 

30 

Sundries. 

567 

40 

" 

22 

Bills  receivable. 

1000 

« 

31 

Merchandise. 

384 

30 

May 

1 

Merchandise. 

317 

Apr. 

22 

Interest. 

16 

48 

" 

17 

Cash. 

424 

•i 

24 

Merchandise. 

846 

51 

July 

1 

Balance. 

852 

May 

17 

" 

387 

25 

3394 

91 

3394 

July 

1 

Balance. 

852 

84 

91 


ANALYSIS. — It  can  readily  be  seen  that  the  debit  side  is  greater;  therefore 
add  that  side  first  and  write  the  sum  as  the  total  or  footing  of  each  side. 
Then  pass  to  the  other  side  of  the  account.  The  sum  of  the  first  column  is 
17,  which  subtracted  from  the  next  higher  number,  21,  ending  with  1,  the 
corresponding  figure  of  the  total,  leaves  4,  which  write  as  the  first  figure  of 
the  balance,  carrying  the  2  to  the  next  column.  (If  the  right-hand  figure  of 
the  sum  of  any  column  is  the  same  as  the  corresponding  figure  of  the  total, 
subtract  it  from  itself,  and  not  from  the  next  higher  number  ending  with  the 
same  figure  ;  or  write  0  in  the  balance  and  carry  the  left-hand  figure  of  the 
sum.)  The  sum  of  the  figures  in  second  column  plus  2  carried  is  11,  which 
subtracted  from  19  leaves  8,  the  second  figure  of  the  balance.  Proceed  in  like 
manner  until  all  the  figures  of  the  balance  are  obtained.  Prove  by  adding 
all  the  numbers,  including  the  balance. 


EXAMPLES. 

85.  Find  the  balances  of  the  following  accounts  : 

(*•)  (*)  (*o 


Dr. 


Or. 


Dr. 


Or, 


Dr. 


Or, 


817.20 

812.20 

237.25 

112.27 

1075. 

375.60 

222.22 

214.13 

900. 

218.36 

2318.42 

218.24 

427.30 

375. 

800. 

717.49 

812.10 

717.37 

810.75 

412. 

718.24 

648. 

938.40 

244.45 

416.30 

717. 

218.75 

118.75 

4312. 

946.33 

225. 

538.98 

222.48 

719.46 

203.13 

108.75 

MULTIPLICATION 


36.  Multiplication  is  the  operation  of  taking  one  number 
as  many  times  as  there  are  units  in  another. 

The  number  taken  or  multiplied  is  called  the  Multiplicand.  The  number 
which  indicates  how  many  times  the  multiplicand  is  taken  or  multiplied,  is 
called  the  Multiplier.  The  result  obtained  is  called  the  Product. 

37.  The  sign  of  multiplication  is    x ,  and  is  read  times,  or 
multiplied  by. 

Thus,  5  x  4=20,  is  read  5  times  4  equals  20,  or  5  multiplied  by  4  equals  20. 

38.  Multiplication  Table. 


21 


9  lO:il! 


18  20  22  24 

27j  30  33  36 

33  40  44  48 

35J  40;  45!  501  55  60 

42j  48j  54|  60  66  72 

49  56  631  70:  77|  84 

56  641  72!  80;  88;  96 

63|  72J  8l!  90j  99108 


70'  80 


a 

0^1 


84!  96108:120132144 


40 


72 


90.100110190 


13  14  15  10  17  18J19  20  31  22  23  34  35 


40 


26;  28  301  32 

39|  42,  45|  48J  51 1  54J  57 
52  56  60j  64,  68,  72,  76  8€ 
65!  70!  75!  80|  85  90!  95100 
78 
91 

104 112J120J128jl36!l44'152J160 
117  126 135 144 153162 17l|l80 
130:i40'l50 160  170 180190200 


84|  90  96102108114120 
98 105112;119 126133140 


99!llO;i21;132|l43  154 165 176  187 198,209  220 
156168 180:198  204  216  228:240 


78j  911041171130,143156169182 195^208  221 


234247,260 


42 


.0611 
26132138144150 
47 154' 161 168 175 
.68  176 184 192  200 
189198207216225 
21012202301240250 
231242'253264275 
2264276288300 
273286,299312325 
294'303  322,336  350 


84  98112126|l40i154168l82196210224238252'266j280 

„  „  90 105 120 135J150165 180195  210^225!240!255|270|285;300  315|33o|345,360  375 
64'  80  96112128144J160176l192l208l224i240'256272288304320336352368384400 
68'  85 102 119 136 153 170;187'204I221;238  255  2721289J306  323  340  357J374  391 408  425 
72:  90108'l26144162180198216'234|252270288|306|324342360  378|396  414  432  450 


•4  oon  or  if  i ' 


761  95,114 133 152 171J190  209  228  247|266|285,304|323|342  361  380 
80!lOO  120 140 160180  200  220240  260:280  300'320  340  860J880I40C 


456475 


84105126147 

88110:i32154176|l{ 

92115!138161 


420:4401460480500 
1(58  189  210  231  252  273  294  315  336  357  378  399  420j44l|462|483|504  525 


1220  242  264  286 


374|396i415 


440462484(506|L 


!550 


264  288,312  336  360,384  408  482  456,480  504|528;552  576  600 
75 100125150175  200J225|250  275  300  325  350J375  400!425  450J475  500  525  550  575  600  625 

1|    3     3     4|    5|    6|    7     8     9110111313114115161718,1913013133333435 


Art.  39.] 


MULTIPLICA  TION. 


23 


39.  To  find  the  product  of  two  numbers  when  the 
multiplier  does  not  exceed  12. 

Ex.    Multiply  456  by  7. 

OPERATION.  ANALYSIS.— 7  times  6  units  are  42  units=4  tens  and  2  units. 

456  Write  the  2  units  under  the  figure  of  the  multiplier  (the  column 

7  of  units)  and  add  the  4  tens  to  the  next  product  (the  column  of 

tens).  7  times  5  tens  are  35  tens,  plus  4  tens  from  the  preced- 
ing product,  are  39  tens  =  3  hundreds  and  9  tens.  Write  the 
9  tens  under  the  column  of  tens,  and  add  the  3  hundreds  to  the  next  product 
(the  column  of  hundreds).  7  times  4  hundreds  are  28  hundreds,  plus  3  hun- 
dreds from  the  preceding  product  are  31  hundreds  =  3  thousands  and  1  hun- 
dred. Write  the  1  hundred  under  the  column  of  hundreds  and  the  3  thousands 
in  the  column  of  thousands. 

40.  RULE. — Commencing  at  the  right,  multiply  each 
figure  of  the  multiplicand  by  the  multiplier,  writing  the 
result  and  carrying  as  in  addition. 


EXAMPLES. 


41 

.  Multiply 

Multiply 

1. 

23456 

by 

?; 

by 

8. 

8. 

789123 

by 

2; 

by 

3. 

2. 

37804 

by 

9; 

by 

6. 

9. 

123567 

by 

4; 

by 

5. 

3. 

24687 

by 

2; 

by 

4. 

10. 

781693 

by 

6; 

by 

9. 

4- 

36925 

by 

3; 

by 

8. 

11. 

417009 

by 

8; 

by 

7. 

5. 

48716 

by 

5; 

by 

9. 

12. 

509048 

by 

8; 

by 

7. 

6. 

90809 

by 

9; 

by 

8. 

13. 

637485 

by 

6; 

by 

9. 

7. 

26048 

by 

5; 

by 

7. 

U. 

748596 

by 

?; 

by 

5. 

15.  There   are   5280  feet   in   one   mile.      How  many  feet  in 
11  miles  ? 

16.  There  are  4  gills  in  one  pint,  2  pints  in  one  quart,  and 
4  quarts  in  one  gallon.     How  many  gills  in  63  gallons  ? 

17.  There  are  2  pints  in  one  quart,  8  quarts  in  one  peck,  and 
4  pecks  in  one  bushel.     How  many  pints  in  379  bushels  ? 

18.  There  are  12  inches  in  one  foot,  and  3  feet  in  one  yard. 
How  many  inches  in  1760  yards  ? 

19.  In  one  gross  there  are  12  dozen,  and  in  one  dozen  12  units. 
Find  the  value  of  45  gross  lead-pencils  at  3  cents  each. 

20.  How  many  buttons  on  6  dozen  pair  of  shoes,  if  there  are 
9  buttons  on  each  shoe  ? 


24  MULTIPLICATION.  [Art.  42. 

42.  To  find  the  product   of  two  numbers,  when  the 
multiplier  is  more  than  12. 

Ex.    Multiply  456  by  237. 

OPERATION.  ANALYSIS. — Write  the   multiplier  under  the  multipli- 

456  cand  so  that  their  right-hand  figures  are  in  the  same  ver- 

237  tical  line.     Since  the  multiplier  consists  of  7  units,  3  tens, 

and  2  hundreds,  the  multiplicand  is  repeated  or  multiplied 

by  7,  by  30,  and  by  200.     7  times  456  is  3192,  the  first  par- 

1368  tial  product  ;    30  times  456  is  13680,    the   second   partial 

912  product;  200  times  456  is  91200,  the  third  partial  product. 

The  sum  of  these  partial  products  is  108072.  the  entire 

product.      In  practice,   the  ciphers  are  omitted.      In  the 

operation,  observe  that  the  first   or   right-hand   figure  of 

each  partial  product  is  directly  under  the  figure  of  the  multiplier  used. 

43.  RULE. — Write  the  multiplier  under  the   multipli- 
cand so  that  their  right-hand  figures  are   in  the  same 
vertical  line. 

Multiply  the  multiplicand  by  each  significant  figure  of 
the  multiplier,  writing  the  first  or  right-hand  figure  of  each 
partial  product  under  the  figure  of  the  multiplier  used. 

Add  the  partial  products.  The  sum  will  be  the  desired 
product. 


EXAM  PLES. 

44.     Multiply  Multiply 

1.  1728  by  37  ;  by  481.  9.  23456  by  294 

2.  2893  by  26  ;  by  506.  10.  40607  by  144 

3.  3904  by  18  ;  by  624.  11.  32738  by  176 

4.  5107  by  41 ;  by  375.  12.  91609  by  201 
6.  6079  by  59  ;  by  208.  13.  24135  by  345 

6.  8125  by  67  ;  by  567.  14.  38246  by  678 

7.  9236  by  78  ;  by  781.  16.  94538  by  987 

8.  7438  by  89  ;  by  936.  16.  10908  by  406 


by  3742. 

by  4803. 
by  5964. 
by  6075. 
by  7186. 
by  8297. 
by  9410. 
by  2465. 


17.  How  many  hours  in  the  month  of  January  ? 

18.  How  many  minutes  in  the  month  of  April  ? 

19.  How  many  seconds  in  the  month  of  February,  1899  ? 

20.  Find  the  cost  of  375  barrels  pork  at  $14  per  barrel. 

21.  There  are  5280  feet  in  one  mile.     How  many  feet  in  96 
miles  ?     In  208  miles  ? 


Art.  44.]  MULTIPLICATION.  25 

22.  How  many  pounds  in  471  bushels  corn,,  if  there  are  5G 
pounds  in  one  bushel  ? 

23.  In  a  bushel  of  timothy  seed,  there  are  45  pounds.     How 
many  pounds  in  2367  bushels  ?     In  3416  bushels  ? 

24.  How  many  shoes   in  24   boxes,  if   each  box  contains  12 
pair  ? 

25.  A   certain   building   has  192  windows,  and  each  window 
contains  24  panes  of  glass.     How  many  panes  in  all  ? 

26.  How  many  feet  of  wire  will  be  required  to  fence  a  field 
209  feet  square,  the  fence  being  6  wires  high  and  on  all  sides  of 
the  field  ? 

45.  To  find  the  product  of  two  numbers  when  there 
are  ciphers  at  the  right   of  the   significant  figures  (6) 
of  one  or  both. 

Ex.    Multiply  37600  by  47000. 

OPERATION.  ANALYSIS. — Write  the  numbers  so  that  the  right-hand 

37600  significant  figures  are  in  the  same  vertical  line.    37600 

47000  =  376  x  100,  and  47000  =  47  x  1000.     Since  the  product 

of  two  or  more  numbers  is  the  same   in  whatever  order 

they  are  multiplied,  multiply  376  by  47,  and  their  product 

1504     by  100000  (100  x  1000),  by  annexing  5  (3  +  2)  ciphers  to 

1767200000      the  right 

46.  RULE. —  Write  the  numbers  so  that  their  right-hand 
significant  figures  are  in  the  same  vertical  line.     Multiply 
the  significant  figures  together  as  if  there  were  no  ciphers, 
and  to  their  product  annex  as  many  ciphers  as  are  found 
on  the  right  of  both  numbers. 

EXAM  PLES. 

47.  Multiply  Multiply 

1.  3600  by  40  ;  by  300.  9.  48400  by  200  ;  by  1400. 

2.  1728  by  80  ;  by  500.  10.  37000  by  500  ;  by  2500. 

3.  3456  by  70  ;  by  420.  11.  12345  by  600  ;  by  3600. 

4.  3710  by  50  ;  by  360.  12.  28000  by  420  ;  by  4700. 

5.  4000  by  30  ;  by  800.  13.  19700  by  340  ;  by  5800. 

6.  2800  by  90  ;  by  370.  14.  14320  by  560  ;  by  6900. 

7.  1360  by  60  ;  by  200.  15.  84000  by  800  ;  by  7320. 

8.  4200  by  20  ;  by  500.  16.  96000  by  900  ;  by  4800. 


26  MULTIPLICATION.  [.\rt.48- 

SHORT     METHODS.* 

48.  To  multiply  any  number  of  two  figures  by  n. 

49.  RULE. — Place  the  sum  of  its  digits  between  them 
when  the  sum   is  less   than    10.      When  the  sum  is  10  or 
more  than   10,  write  Us  right-hand  figure  in  the  second 
place  and  carry  one  to  the  left-hand  figure  of  the  multi- 
plicand. 

EXAMPLES. 

50.  1.  Multiply  34  by  11. 

ANALYSIS. — 3  +  4  =  7,  which  placed  between   3    and    4    produces    the 
product  374. 

2.  Multiply  68  by  11. 

ANALYSIS. — 6  +  8  =  14.     Write  4  in   the  second   place  and   carry  1  to 
the  6,  the  left-hand  figure  of  the  multiplicand  producing  the  product  748. 

3.  Multiply  the  following  numbers  by  11 :  24,  16,  18,  32,  43, 

33,  72,  81,  37,  44,  92,  87,  93,  64,  35,  36,  47,  17,  19,  48,  and  57. 

51.  To  multiply  any  number  by  n. 

52.  EULE. —  Write  the   1st  j*ight-hand  figure,   add  the 
1st  and  2nd,  the  2nd  and  3rd,  and  so  on ;  finally  write 
the  left-hand  figure,  carrying  as  usual. 

EXAMPLES. 

53.  1.  Multiply  783742  by  11.  Ans.  8621162, 
ANALYSIS. — Write  the  right-hand  figure  2 ;  for  the  remaining  figures  of 

the  product,  add  2  to  4,  4  to  7,  7  to  3,  3  to  8,  8  to  7,  and  write  the  left-hand 
figure,  carrying  when  necessary. 

2.  Multiply  the  following  numbers  by  11  :  245,  346,  325,  416, 
784,  517,  875,  918,  4218,  7324,  7218,  1728,  4375,  and  8376. 

54.  To  multiply  by  any  number  of  two  figures  ending 
with  I. 

55.  RULE. — Multiply    by    the  tens  of   the    multiplier, 
writing  the  product  under  the  multiplicand  one  place  to  the 
left,  and  add.     Or, 

*  It  is  suggested  that  these  short  methods  be  studied  in  connection  with  the  more  advanced 
work— one  method  with  each  lesson;  or  they  may  be  presented  to  the  student,  one  at  a  time, 
with  the  daily  drill  exercises  on  the  fundamental  rules. 


Art.  55.]  SHORT   METHODS.  27 

Write  as  the  first  figure  of  the  product  the  unit  figure  of 
the  multiplicand;  multiply  each  figure  of  the  multipli- 
cand by  the  tens  of  the  multiplier,  and  at  the  same  time, 
add  mentally  to  each  product  the  figure  to  the  left  of  the 
one  multiplied,  carrying  as  usual. 

EXAMPLES. 

56.  1.  Multiply  456  by  61. 

1ST  OPERATION.      2ND  OPERATION. 

ANALYSIS,  SND  METHOD.— Write  6  in  the 
product,    6  x  6  +  5  =  41.    Write  1  and  carry 

2736  61  4.    6  x  5  +  4  (carried)  +4  =  38.    Write  8  and 

27816  27816  carry  3.    6x4  +  3  (carried)  =  27. 

Multiply  Multiply 

2.  864  by  61 ;  by  41.  5.  2345  by  121 ;  by  111. 

3.  717  by  31 ;  by  71.  6.  7416  by  51 ;  by  81. 

4.  447  by  21 ;  by  81.  7.  8324  by  41 ;  by  21. 

NOTE. — The  first  method  may  be  used  with  the  following  multipliers,  by 
placing  the  products  two  places  to  the  left. 

Multiply  Multiply 

8.  375  by  301 ;  by  401.  11.     483  by  701 ;  by  801. 

9.  425  by  201 ;  by  101.  12.     376  by  201 ;  by  901. 
10.     469  by  601 ;  by  501.  18.     875  by  301 ;  by  401. 

57.  To  multiply  by  any  number  between  12  and  20. 

58.  EULE. — Multiply    ~by  the  units  of  the  multiplier, 
writing  the  product  under  the  multiplicand  one  place  to 
the  right,  and  add.     Or, 

Multiply  the  units  of  the  multiplicand  by  the  units  of 
the  multiplier,  write  the  units  of  the  product,  and  carry 
the  tens,  if  any,  to  the  next  product ;  multiply  the  remain- 
ing figures  of  the  multiplicand  by  the  units  of  the  multi- 
plier, and  at  the  same  time  add  mentally  to  each  product 
the  figure  to  the  right  of  tfie  one  multiplied,  carrying  as 
usual ;  finally,  to  the  left-hand  figure  of  the  multiplicand, 
add  the  number  to  be  carried,  if  any,  and  write  the  result. 


28 


UL  TIP LIC A  TION. 


[Art.  59. 


EXAM  PLES. 


59.     1.  Multiply  456  by  18. 

1ST  OPERATION.  2ND  OPERATION. 

456  456 

3648  18 

8208  8208 


ANALYSIS,  SND  METHOD.  —  8  x  6  =  48. 
Write  8  and  carry  4.  8x5  +  4  (carried)  +  6  = 
50.  Write  0  and  carry  5.  8x4  +  5  (carried) 
+  5  =  42.  Write  2  and  carry  4.  4  +  4  =  8. 


Multiply 

2.  785  by  13 

3.  378  by  14 

4.  522  by  15 


by  17. 
by  16. 
by  19. 


Multiply 

6.  1234  by  14 

7.  2345  by  16 

8.  3456  by  19 


by  16. 
by  18. 
by  13. 


5.     376  by  18  ;  by  16. 


9.     7891  by  17  ;  by  15. 


NOTE. — The  first  method  may  be  used  with  the  following  multipliers  by 
placing  the  products  two  places  to  the  right. 


Multiply 

10.  875  by  101  ;  by  108. 

11.  936  by  102  ;  by  103. 

12.  877  by  104  ;  by  106. 
18.     736  by  105  ;  by  109. 


Multiply 

14.  147  by  108  ;  by  101. 

15.  385  by  104  ;  by  107. 

16.  783  by  105  ;  by  103. 

17.  546  by  107  ;  by  106. 


60.  To  multiply  by  any  number  ending  with  9. 

61.  RULE. — Multiply  by  1  more  than  the  given  multi- 
plier, and  from  the  result  subtract  the  multiplicand. 


EXAMPLES. 


62.     1.   Multiply  387  by  49. 


OPERATION. 


387  product  by     1 
19350         "        "    50 

49  (Subtracted  downwards.) 


18963 


Multiply 

2.  76  by  49  ;  by  39. 

3.  87  by  29  ;  by  99. 

4.  45  by  59  ;  by  69. 


Multiply 

5.     312  by  19  ;  by  89. 
fc.     427  by  39  ;  by  79. 

7.     825  by  29  ;  by  69. 


Art.  63.]  SHORT    METHODS.  29 

63.  To  multiply  by  any  multiple  of  9  less  than  90. 

64.  RULE.  —  Multiply  by  the  multiple  of  ten  next  higher 
than  the  given  multiplier,  and  from  the  result  subtract  one- 
tenth  of  itself. 

EXAM  PLES. 

65.  1.   Multiply  785  by  63. 

OPERATION. 

785 

70  ANALYSIS.—  63  =  70-7.     785  x  70  =  54950. 

54950  product  bv  70          Divide  5495°  by  10  by  placing  its  digits  one 

place  to  the  right.     54950-5495  =  48455. 

7 


49455  63 

Multiply  Multiply 

2.     67  by  18  ;  by  27.  6.     345  by  36  ;  by  45. 

8.     34  by  36  ;  by  45.  7.     567  by  18  ;  by  72. 

4.  77  by  54  ;  by  63.  8.     518  by  27  ;  by  63. 

5.  84  by  72  ;  by  81.  9.     724  by  54  ;  by  81. 

66.  To  multiply  by  25. 

67.  RULE.  —  Add  two   ciphers  and   divide  the  result  by 
4-     Or, 

Divide  the  number  by  4  /  if  there  is  no  remainder,  add 
two  ciphers  ;  if  there  is  a  remainder  of  1,  add  25  ;  of  2, 
add  50;  of  3,  add  7  5. 

EXAMPLES. 

68.  1.  Multiply  446  by  25. 

ANALYSIS.—  Since  25  is  equal  to  100  divided  by  4,  multi- 
plying by  100  and  dividing  the  result  by  4,  is  the  same  as 
11150  multiplying  by  25. 

2.  Multiply  the  following  numbers  by  25  :—  24,  36,  37,  49,  62, 
387,  448,  512,  746,  424,  817,  9S7,  544,  717,  318,  324,  256,  556, 
9224,  8378,  5280,  1728,  5648. 


30  MULTIPLICATION.  [Art,  69. 

69.  To  multiply  by  any  number  one  part  of  which  is 
a  factor  of  another  part. 

EXAMPLES. 

70.  1.  Multiply  576  by  287.         2.  Multiply  567  by  936. 

OPERATION.  OPERATION. 

576  567 

287  936 


4032  product  by   7.       5103   product  by  9. 
16128    "   "   28(4x7).   20412    "   "  36(4x9). 
530712   "'   "  287.       530712    "   "  936. 

Multiply  Multiply 

3.  227  by  369  ;  by  427.  8.     932  by  183  ;  by  927. 

4.  516  by  246  ;  by  568.  9.     718  by  284  ;  by  832. 

5.  344  by  126  ;  by  124.  10.     529  by  546  ;  by  756. 

6.  728  by  426  ;  by  189.  11.     638  by  217  ;  by  618. 

7.  325  by  147  ;  by  273.  12.     435  by  248  ;  by  428. 

71.  To  multiply  by  any  number  near  and  less  than 
100,  1000,  etc. 

72.  The  Complement  of  a  number  is  the  difference  between 
the  number  and  the  unit  of  the  next  higher  order. 

73.  RULE. — Add  to  the  multiplicand  as  many  ciphers 
as  there  are  ciphers  in  the  unit  next  higher  than  the  mul- 
tiplier, and  from  the  result  subtract  the  product  obtained 
by  multiplying  the  multiplicand  by  the  complement  of  the 
multiplier. 

EXAMPLES. 

74.  1.  Multiply  456  by  98. 

OPERATION. 

45600  product  by  100.' 

912    "   "  _2. 

44688    "   "  ~98. 

Multiply  Multiply 

2.  77  by  99  ;  by  93.  6.  387  by  93  ;  by  999. 

8.  84  by  98  ;  by  95.  6.  416  by  95  ;  by  994. 
4.  72  by  94  ;  by  96.  7.  528  by  93  ;  by  992. 


Art.   75.] 


SHORT     METHODS. 


31 


CROSS  MULTIPLICATION. 

75.   Cross    Multiplication     depends    upon    the    following 
principles : 


Units        multiplied  by  units 


Tens 

units 

Units 

tens 

Hundreds          " 

units 

Tens 

tens 

Units 

hundreds 

Thousands        "          " 

units 

Hundreds 

tens 

Tens 

hundreds 

Units 

thousands 

Ten-thousands  "          " 

units 

Thousands        "          " 

tens 

Hundreds           "          " 

hundreds 

Tens 

thousands 

Units 

ten-thousands 

Etc.,  etc. 

Ex.     Multiply  68  by  74. 


produce  units 
"       tens. 

hundreds. 
"        thousands. 

"        ten-thousands. 

Am.  5032. 


OPERATION. 


68 

74 


4x6  +  3  (carried)  +  7 


8  =  3 

8  =  8 


5032 


7x6  +  8  (carried)  =  50 


Ex.     Multiply  579  by  42. 


Am.  24318. 


OPERATION. 

579 
42 


ANALYSIS. 


24318 


2x9  =  1 

2x7  +  1  (carried)  +4x9  =  5 
2x5  +  5  (carried)  +4x7  =  4 


Ex. 


4x5+4  (carried)  =  24 

Ans.  197316. 


Multiply  567  by  348. 

OPERATION.  ANALYSIS. 

567  8  x  5  =  40  8  x  6  =  48  8  x  7  =  56 

348  4x5  =  20  4x6  =  24  4x7  =  28 

197316  3x5  =  15  3x6  =  18  3x7  =  21 . 

7  3  1  ~  6 


19 


32  MULTIPLICATION.  [Art.  7G. 

76.   To  multiply    together  numbers  of  two    figures 
each,  whose  units  are  alike. 

Ex.    Multiply  76  by  46.  Ans.  3496. 


OPERATION.  ANALYSIS. 


6  6x6  =  3 


46 


*4  j-  6x11  +  3  (carried)  =  6 


6 


9 


3496 

4x    7  +  6  (carried)  =  34 


Ex.    Multiply  135  by  65.  Ans.  8775. 

ANALYSIS. 

5x5  =  2 
1-5x19  +  2  (carried)  =  9 


OPERATION.  ANALYSIS. 

135  5x5  = 

65  5  x  13 


8775  5x    6 


6x13  +  9  (carried)  = 


77.  RULE. — Multiply  units  by  units  for  the  first  figure 
of  the  product,  the  sum  of  the  tens  by  units  for  the  second 
figure,  and  tens  by  tens  for  the  third  figure,  carrying  when 
necessary. 

EXAMPLES. 

78.  Multiply 

1.  56  by  56  ;  72  by  32  ;  94  by  44. 

2.  65  by  75  ;  87  by  37  ;  46  by  36. 

3.  99  by  49  ;  85  by  75  ;  34  by  24. 

4.  47  by  37  ;  67  by  57  ;  85  by  45. 

5.  125  by  65  ;  126  by  36  ;  154  by  84. 

6.  76  by  76  ;  36  by  36  ;  114  by  114. 

79.  To    multiply    together   numbers   of  two   figures 
each,  whose  tens  are  alike. 

Ex.    Multiply  87  by  85.  Ans.  7395. 


OPEBATION.  ANALYSIS. 

87  5x7  =  3 

85  8  x 


7395  »  X 

8  x    8  +  9  =  7 


9 


Art.  79.]  SHORT    METHODS.  33 

Ex.    Multiply  127  by  122.  Ans.  15494. 

OPERATION.  ANALYSIS. 

127  2x7=1 

122  12  x  2  ) 

1^494  12x7J12><    9+    1  =  10 

12  x  12  +  10  =  15     4 

80.  RULE. — Multiply  units  by  units  for  the  first  figure 
of  the  product,  the  sum  of  the  units  by  tens  for  the  second 
figure,  and  tens  by  tens  for  the  remaining  figures,  carrying 
when  necessary. 

EXAMPLES. 

81.  Multiply 

1.  87  by  82  ;  81  by  87  ;  65  by  63. 

2.  47  by  44  ;  56  by  52  ;  58  by  57. 
8.     73  by  76  ;  79  by  75  ;  68  by  63. 

4.  44  by  43  ;  52  by  55  ;  67  by  63. 

5.  116  by  117  ;  107  by  105  ;  125  by  122. 

82.  To   multiply  together  two   numbers  whose  tens 
are  alike,  and  the  sum  of  whose  units  is  ten. 

83.  RULE. — Multiply  the    units    together   for    the    two 
right-hand  figures  of  the  product,  one  of  the  tens  by  1  more 
than  itself  for  the  remaining  figures. 


EXAMPLES. 

84.     1.  Multiply  76  by  74.  Ans.  5624. 

ANALYSIS.  —  6  x  4  =  24,   the  two  right-hand    figures    of    the  product. 
G  x  7  (6  +  1)  =  42,  the  remaining  figures. 

Multiply  mentally 

2.  24  by  26  ;  85  by  85  ;  128  by  122. 

3.  17  by  13  ;  94  by  96  ;  112  by  118. 

4.  34  by  36  ;  37  by  33  ;  104  by  106. 

5.  25  by  25  ;  43  by  47  ;  143  by  147. 

6.  35  by  35  ;  56  by  54  ;  152  by  158. 


34  MULTIPLICATION.  [Art.  85. 

85.  To  multiply  by  means  of  complements  (72). 

Ex.    Multiply  991  by  996. 

OPERATION.  ALGEBRAIC  MULTIPLICATION. 

991  ..  9  991  =•  1000  —  9  ) 

996..  4  996  =  =-  13 


987036  1000  x  1000  —  9  x  1000 

___  —  4  x  1000  +  36 
(1000  —  13)  x  1000     +  36 

ANALYSIS.  —  From  the  above  algebraic  multiplication,  it  is  observed  : 
1st,  that  as  many  of  the  right-hand  figures  as  there  are  ciphers  in  the  unit  of 
comparison  may  be  obtained  by  multiplying  the  complements  together  ;  2nd, 
that  the  second  part  of  the  result  is  equivalent  to  the  sum  of  the  numbers  less 
the  unit  of  comparison  multiplied  by  that  unit. 

The  sum  of  the  numbers  less  the  unit  of  comparison  may  be  obtained  by 
adding  the  numbers  and  omitting  the  1  at  the  left-hand,  or  by  subtracting 
either  complement  from  the  opposite  number.  Thus,  991  —  4  =  987. 

86.  RULE.  —  From   either  number  subtract  the   comple- 
ment of  the  other,  and  to  the  right  of  the  remainder  write 
the  product  of  the  complements. 

NOTES.  —  1.  When  there  are  less  figures  in  the  product  of  the  comple- 
ments than  ciphers  in  the  unit  of  comparison,  write  ciphers  in  the  result  to 
supply  the  deficiency. 

2.  When  there  are  more  figures  in  the  product  of  the  complements  than 
ciphers  in  the  unit  of  comparison,  add  the  excess   on  the  left-hand  to  the 
second  part  of  the  result. 

3.  After  practice,  the  complements  may  be  omitted  in  the  operation. 

EXAMPLES. 

87.  1.  Multiply  88  by  95  ;  975  by  993  ;  9999  by  9999. 

(«-)  (*.)  ('-) 

88..  12  775..  225  9999...  1 

95.  ..5  993  ....  7  9999...  1 


8360  769575  99980001 

Multiply  Multiply 

2.  97  by  99  ;  by  94.  8.  993  by  992  ;  by  994 

3.  88  by  91  ;  by  95.  9.  990  by  991  ;  by  988. 

4.  89  by  93  ;  by  96.  10.  982  by  994  ;  by  995. 

5.  75  by  97  ;  by  98.  11.  925  by  996  ;  by  994. 

6.  92  by  98  ;  by  93.  12.  875  by  992  ;  by  993. 

7.  86  by  94  ;  by  95.  13.  847  by  990  ;  by  988. 


Art,  88.]  SHORT   METHODS.  35 

88.  To  multiply  together  two  numbers  of  the   same 
number  of  figures  over  and  near  100,  1000,  etc. 

Ex.    Multiply  116  by  103. 

OPERATION.  ALGEBRAIC  MULTIPLICATION. 

116  116  =  100  +  16  I  -900+19 

103  =  100+  af811111"200  +  9 

100  x  100  +  16  x  100 

H948  +    3  x  100  +  48 

(100  +  19)  x  100  +  48 

89.  RULE. — From  the  sum  of  the  numbers  subtract  the 
unit  of  comparison,  and  to  the  right  of  the  result  write  the 
product  of  the  excesses.     (See  Notes  to  Art.  86.) 

EXAMPLES. 

90.  Multiply  Multiply 

1.  112  by  106  ;  by  111.  5.  145  by  107  ;  by  112. 

2.  102  by  103  ;  by  104.  6.  176  by  111 ;  by  108. 

3.  122  by  108  ;  by  105.  7.  1004  by  1006  ;  by  1007. 

4.  116  by  107  ;  by  112.  8.  1125  by  1008  ;  by  1012. 

91.  To  multiply  together  two  numbers,  one  of  which 
is  more  and  the  other  less  than  100,  1000,  etc., 

Ex.    Multiply  109  by  97. 

OPERATION.  ALGEBRAIC  MULTIPLICATION. 

109         9  excess.  109  = 

97         3  complement.  97  = 

10600  ,  100  x  100  +  9  x  100 

f  Product  of  excess  _  3  x  100  -  27 

-  (  and  complement. 

10573  (100  +  6)  x  100  -  27 

92.  RULE. — Multiply  the  sum  of  the  nuinbers  less  the 
unit  of  comparison  ~by  that  unit,  and  from  the  product 
subtract  the  product  of  the  excess  and  complement. 

EXAMPLES. 

93.  Multiply  Multiply 

1.  107  by  97  ;  by  95.  5.  1005  by  91 ;  by  93. 

2.  112  by  96  ;  by  92.  6.  1007  by  95  ;  by  97. 

3.  116  by  94  ;  by  98.  7.  1012  by  99  ;  by  92. 

4.  108  by  91 ;  by  99.  8.  1018  by  94  ;  by  96. 


DIVISION. 


94.  Division  is  the  operation  of  finding  how  many  times 
one  number  is  contained  in  another. 

The  number  divided  is  called  the  dividend.  The  number  by  which  it  is 
divided  is  called  the  divisor.  The  result  obtained  is  called  the  quotient. 
The  part  of  the  dividend  which  remains  after  the  operation  is  completed  is 
called  the  remainder. 

95.  The  sign  of  division  is  -±-  and  is  read  divided  by. 
Thus,  16  -f-  2  =  8  is  read,  sixteen  divided  by  two  equals  eight. 

96.  To  divide  when  the  divisor  does  not  exceed  12. 

NOTE. — When  the  work  is  performed  mentally,  as  in  the  following  opera- 
tion, the  process  is  called  Short  Division. 

Ex.    Divide  1859  by  4. 

OPEBATION.  ANALYSIS.— Write  the  divisor  at  the  left  of  the  dividend, 

4  )  1859  as  in  the  operation,  and  begin  to  divide  at  the  left.  4  is 

4645.  n°k  contained  in  1  thousand,  the  highest  order  of  the 
dividend,  therefore,  divide  18  hundreds  by  4.  4  is  con- 
tained in  18  (hundreds),  4  (hundred)  times,  and  2  hundreds 
remain.  Write  the  9  hundred  under  the  line  in  hundreds'  place,  and  reduce 
the  2  hundreds  remaining  to  tens,  making  20  tens,  which  added  to  the  5  tens 
of  the  dividend,  make  25  tens.  4  is  contained  in  25  (tens),  6  (tens)  times  and 
1  ten  remains.  Write  the  6  tens  under  the  line  in  tens'  place,  and  reduce  the 
1  ten  remaining  to  units,  making  10  units,  which  added  to  the  9  units  of  the 
dividend,  make  19  units.  4  is  contained  in  19  (units),  4  (units)  times,  and  3 
units  remain.  Write  the  4  units  in  units'  place,  and  write  the  remainder 
over  the  divisor,  with  a  line  between  them  in  the  form  of  a  fraction,  thus,  f 
(three-fourths).  The  complete  result  is  464f . 

Observe  that  each  quotient  figure  is  placed  directly  under  the  last  figure 
of  the  dividend  used. 

In  practice,  do  not  think  of  explanations,  etc. ;  but,  think  only  of  the  par- 
tial dividends  and  quotient  figures.  Thus,  in  the  above  example,  say  or  think, 
4  into  18  4  times,  into  25  6  times,  into  19  4  times,  etc. 


Art.  97.]  DIVISION.  37 

97.  RULE.— Write  the  divisor  at  the  left  of  the  dividend 
with  a  curved  line  between  them. 

Beginning  at  the  left,  divide  each  figure  of  the  dividend 
by  the  divisor,  and  place  the  quotient  beneath  the  figure 
divided.  Whenever  a  remainder  occurs,  prefix  it  to  the 
following  figure  of  the  dividend,  and  divide  as  before. 

Continue  the  operation  until  all  the  figures  of  the 
dividend  have  been  divided,  and  place  the  remainder,  if 
any,  over  the  divisor  at  the  right  of  the  quotient. 

98.  PROOF. — Multiply  the  quotient  by  the  divisor,  and  to 
the  product  add  the  remainder.    If  the  result  equals  the 
dividend,  the  work  is  probably  correct. 


99. 

1. 

2. 
3. 

4. 

5. 
6. 
7. 
8. 
9. 


21.  In  one  square  yard  there  are  9  square  feet.     How  many 
square  yards  in  41652  square  feet  ? 

22.  There  are  12  pence  in  one  shilling.     How  many  shillings 
in  124656  pence  ? 

23.  In  a  barrel  containing  1068  eggs,  how  many  dozen  ?   What 
is  their  value  at  23  cents  per  dozen  ? 

24.  In  one  foot  there  are  12  inches.      How   many  feet   in 
63360  inches  ? 

25.  There  are  2  pints  in  one  quart,  and  4  quarts  in  one  gallon. 
How  many  gallons  in  160048  pints  ? 

26.  There  are  8  quarts  in  one  peck,  and  4  pecks  in  one  bushel. 
How  many  bushels  in  349056  quarts  ? 


EXAM 

PLES 

• 

Divide 

Divide 

78912348  by  2 

;  by  3. 

11. 

103050709 

by 

2; 

by 

5. 

97652464  by  4  ; 

by  6. 

]2. 

214161810 

by 

3; 

by 

6. 

16327620  by  5 

i 

by  6 

. 

13. 

425262728 

by 

4; 

by 

7. 

78070804  by  4 

; 

by  7 

. 

14. 

123456789 

by 

3; 

by 

8. 

12345678  by  6 

j 

by  9 

. 

15. 

246801234 

by 

6  ; 

by 

11. 

988654320  by  5 

;by 

8. 

16. 

789123650 

by 

7; 

by 

10. 

234568836  by 

4 

;by 

9. 

17. 

287236450 

by 

5; 

by 

12. 

357212254  by 

2 

;by 

7. 

18. 

176111888 

by 

6; 

by 

11. 

246886425  by 

5 

;by 

9. 

19. 

1010101010  by  7 

;  by  9. 

217181916  by 

7 

;by 

9. 

20. 

200200200 

by 

8; 

by 

12. 

38  DIVISION.  [Art.   100. 

100.  To  divide  by  any  divisor  greater  than  12. 

NOTE. — When  the  work  is  all  written,  as  in  the  following  operation,  the 
process  is  called  Long  Division. 

Ex.    Divide  13218  by  43. 

•OPERATION.  ANALYSIS.— Since  43  is  not  contained  in  13 

Divisor.  Dividend.  Quotient.          (thousands),   we  take   132   (hundreds)   for   the 
43  )  13218  (  307i-J          first  partial  dividend.     43  is  contained  in  132 
129  (hundreds),   3  (hundred)  times.     43  x  3  (hun- 

TT7  dreds)  =  129   (hundreds),   which  write    under 

the  132   (hundreds),    and  subtract.      The  re- 
mainder is  3  (hundreds),  to  which  annex  the  1 
17  Bemainder.        (ten)  of  the  dividend,  and  the  second  partial 
dividend  is  31  (tens).    43  is  not  contained  in  31 

(tens),  therefore  write  0  as  the  next  figure  of  the  quotient.  Annex  to  the 
partial  dividend,  31  (tens),  the  8  (units)  of  the  dividend,  and  the  next  partial 
dividend  is  318  (units).  43  is  contained  in  318  (units),  7  (units)  times.  43  x  7 
(units)  =  301  (units),  which  write  under  the  318  (units)  and  subtract.  The 
remainder  is  17  (units).  Indicate  the  division  of  this  remainder  in  the  form 
of  a  fraction,  thus :  £$,  and  annex  it  to  the  quotient,  producing  307||  for  the 
complete  quotient. 

101.  RULE. —  Write  the  divisor  at  the  left  of  the  dividend, 
with  a  curved  line  between  them. 

Take  for  the  first  partial  dividend  the  least  number  of 
figures  on  the  left  that  will  contain  the  divisor,  and  write 
the  quotient  figures  at  the  i^ight. 

Multiply  the  divisor  by  the  quotient,  ivrite  the  product 
under  the  partial  dividend,  and  subtract.  To  the  remain- 
der, annex  the  next  figure  of  the  dividend,  for  the  second 
partial  dividend. 

Divide  as  before,  and  thus  continue  until  all  the  figures 
of  the  dividend  have  been  used. 

Write  the  remainder,  if  any,  over  the  divisor  in  the  form 
of  a  fraction,  and  annex  it  to  the  quotient.  The  result  will 
be  the  complete  quotient. 

102.  PROOF. — Multiply  the  divisor  by  the  quotient,  and 
to  the  product  add  the  remainder.     If  the  sum  equals  the 
dividend,  the  work  is  probably  correct. 


Art.  103.]  DIVISION.  39 


EXAM  PLES. 

1O3.    Divide  Divide 

1.  307845  by  26  ;  by  143.  11.     8712460  by  73  ;  by  817. 

2.  248916  by  19  ;  by  249.  12.     1428716  by  84  ;  by  365. 
8.     375428  by  38  ;  by  375.  13.     2893429  by  69  ;  by  144. 

4.  481369  by  48  ;  by  116.  14.  7364128  by  14  ;  by  128. 

5.  423706  by  25  ;  by  208.  15.  2125639  by  70  ;  by  320. 

6.  3064028  by  18  ;  by  429.  76?.  3756425  by  64  ;  by  231. 

7.  1289434  by  64  ;  by  567.  17.  4183691  by  36  ;  by  365. 

8.  7090805  by  73  ;  by  432.  18.  3804072  by  96  ;  by  729. 

9.  6321457  by  87  ;  by  618.  19.  1653891  by  33  ;  by  640. 
10.  2304802  by  92  ;  by  729.  20.  2763940  by  95  ;  by  160. 

21.  How  many  days  in  8766  hours  ? 

22.  In  20000  pens,  how  many  gross  ?     (1  gross  =  144.) 

23.  How  many  bushels  in  21674  pounds  of  oats,  if  there  arc 
32  pounds  in  one  bushel  ? 

24-   There  are  56  pounds  in  a  bushel   of   rye.     How   many 
bushels  in  19958  pounds  ? 

25.  There  were  31392893  gallons  of  molasses   imported  into 
the  United  States  in  1885.     How  manys  hogsheads  of  63  gallons 
each  ? 

26.  How  many  cords  in  47164  cubic  feet,  if  there  are  128  cubic- 
feet  in  one  cord  ? 

27.  How  many  miles  in  49164  rods,  if  there  are  320  rods  in 
one  mile  ? 

28.  The  expenditures  of  the  United  States  for  the  year  1880 
were  $287,034,182.     How  much  was  that  per  day  (365  days  in 
the  year)  ? 

29.  During  the  year  1882,  788992  immigrants  arrived  in  the 
United  States.     What  was  the  average  number  per  day  ? 

30.  The  population  of  the  38  States  was  49,371,340  in  1880. 
and  there  are  325  members  in  the  House  of  Representatives.    What 
is  the  average  population  to  each  member  ? 

31.  The    exports    of    cotton    during    the    year    1885    were 
1,889,514,368  pounds.       How  many  bales  averaging  476  pounds 
each  ? 

82.  How  many  rails  18  feet  in  length  would  be  required  for  a 
railroad  51  miles  long  ?     (1  mile  =  5280  feet.) 


40  DIVISION.  [Art.  104. 


SHORT    METHODS    OF    DIVISION. 

104.  Leaving  out  the  Products.  —  In  long  division    the 
process  may  be  shortened  by  the  following  rule  : 

105.  RULE.—  Subtract  the  several  products  from  the  next 
number  greater  ending  with  the  corresponding  figure  in 
the  dividend,  and  carry  each  time  the  left-hand  figure  of 
the  minuend  to  the  next  product. 

NOTE.  —  If  the  right-hand  figure  of  any  product  is  the  same  as  the  corres- 
ponding figure  of  the  dividend,  subtract  it  from  itself,  and  not  from  the  next 
higher  number  ending  with  the  same  figure  ;  or,  write  0  in  the  remainder, 
carrying  the  left-hand  figure  of  the  product. 

Ex.    Divide  42343014  by  973. 


973  ANALYSIS.—  The  first  quotient  figure  is  4,  by 

which  we  multiply.     4  times  3  are  12,  which  sub- 


42343014 

3423 

tracted  from  14  (the  next  number  greater  ending 

with  4)  leaves  2.     Write  2  in  the  remainder  and 
carry   1.    4  times  7  are  28,  1  carried  makes  29, 
7784  which    subtracted    from    33    (the    next    number 

000  greater  ending  with  3)  leaves  4.     Write  4  in  the 

remainder  and   carry  3.      4  times   9   are  36,  3 

carried  makes  39,  which  subtracted  from  42  leaves  3.  Write  3  in  the  remain- 
der and  carry  4.  4  subtracted  from  4  .leaves  0.  Bring  down  3,  the  next 
figure  of  the  dividend.  So  proceed  until  the  division  is  finished. 

NOTE.  —  Perform  any  of  the  examples  in  Art.  1O3  by  this  method. 

106.  To  divide  by  25. 

107.  RULE.  —  Multiply  the  dividend  by  4>  ana  divide  the 
product  by  100  by  cutting  off  two  figures  from  the  right. 

NOTE.  —  To  divide  by  125,  multiply  by  8  and  divide  the  product  by  1000 
by  cutting  off  three  figures  from  the  right. 

Ex.   Divide*  11175  by  25. 

OPERATION. 

ANALYSIS.  —  Since  25  is  one-fourth  of  100,   multiplying 
___  4        by  4  an(j  dividing  by  100,  is  the  same  as  dividing  by  25. 

447.00 

EXAMPLES. 

108.  1.  Divide  the  following  numbers  by  25  :    1175,  1650, 
1700,  2875,  3825,  4950,  3800,  1725,  1775,  1825,  and  2000. 


UNITED     STATES     MONEY. 


109.  United  States  Money  is  the  legal  currency  of  the 
United  States.      It   consists   of  gold,  silver,  nickel,   and  copper 
coins,  treasury  and  national  bank  notes,  gold  and  silver  certificates. 

110.  Legal  Tender. — The  term  legal  tender  is  applied  to 
money  which  may  be  legally  oifered  in  the  payment  of  debts. 

111.  The  unit  of  value  is  the  gold  dollar  of  25.8  grains. 

TABLE. 


10  Mills  =  1  Cent  c.,  ct. 
10  Cents  =  1  Dime  d. 
10  Dimes  =  1  Dollar  $. 
10  Dollars  =  1  Eagle  E. 


NOTES. — 1.  In  business  operations, 
dollars  and  cents  are  principally  used. 
Eagles  and  dimes  are  used  only  as  the 
names  of  coins. 

2.  The  currency  of  the  Canadian 
Provinces  is  nominally  the  same  as  that 
of  the  United  States. 


112.  The  legal  coins 

GOLD. 

Weight 
in  grains. 

1  dollar  piece,  25.8 

2£  dollar  piece,  or  ) 

Quarter-eagle,     J 
3  dollar  piece,  77.4 

5  dollar  piece,  or  )  Ig9 

Half-eagle,          \ 
10  dollar  piece,  or  ) 

Eagle,  t* 

20  dollar  piece,  or  ) 

Double-eagle,       ( 


of  the  United  States  are  as  follows  : 

SILVER. 

Weight. 
Standard  dollar,        412£  grains. 

Half  dollar,  or 


,  or  ) 

>        12*  grams,  or  192.9  grains. 
50  cent  piece,  ) 

Quarter  dollar,  or  )  _. 

>  61  grams,  or  96.45  grams. 
25  cent  piece,       ) 

24  grams,  or  38.58  grams. 


piece 
Dime,  or 

10  cent  piece, 

COPPER  AND  NICKEL. 

5  cent  piece,  5  grams,  or  77.16  grains. 

3  cent  piece,  30  grains. 

1  cent  piece,  48  grains. 


113.  The  gold  and  silver  coins  of  the  United  States  contain  9 
parts  by  weight  of  pure  metal  and  1  part  alloy.  The  alloy  of 
silver  coins  is  copper ;  and  of  gold  coins,  copper,  or  copper  and 
silver.  (The  silver  in  no  case  exceeds  ^  of  the  whole  alloy.) 


42  UNITED     STATES     MONEY.  [Art.  114. 

114.  Gold  Coins  are  a  legal  tender  in  all  payments  at  their 
nominal  value  when  not  below  the  standard  weight  *  provided  by 
law ;  and,  when  reduced  in  weight,  below  said   standard,  are  a 
legal  tender  at  valuation  in  proportion  to  their  actual  weight. 

115.  Standard  Silver  Dollars  are  a  legal  tender  at  their 
nominal   value   for   all   debts   except   where  otherwise  expressly 
stipulated  in  the  contract. 

116.  Silver  Certificates. — Any  holder  of  standard  silver 
dollars  may  deposit  the  same  with  the  Treasurer,  or  any  Assistant 
Treasurer  of  the  United  States,  in  sums  not  less  than  $10,  and 
receive  therefor  certificates,  corresponding  with  the  denominations 
of  United  States  notes  (119).      These  certificates  are  receivable 
for  customs,  taxes,  and  all  public  dues. 

117.  Subsidiary  Coins. — The  present  silver  coins  of  the 
United  States  of  smaller  denominations  than  $1  are  a  legal  tender 
in  all  sums  not  exceeding  $10. 

The  holder  of  any  of  the  silver  coins  of  the  United  States  of  smaller 
denominations  than  $1  may,  on  presentation  of  the  same  in  sums  of  $20,  or 
any  multiple  thereof,  at  the  office  of  the  Treasurer  or  any  Assistant  Treasurer 
of  the  United  States,  receive  therefor  lawful  money  of  the  United  States. 

118.  Minor  Coins. — The  5   and  3   cent  pieces  contain  } 
copper  and  £  nickel.     The   1  cent  piece  contains  95  per  cent, 
copper   arid   5   per  cent,  tin  and  zinc.     These  coins  are  a  legal 
tender  for  any  amount  not  exceeding  twenty-five  cents. 

119.  United   States   Notes    ("Greenbacks")   are  a  legal 
tender  for  all  debts  except  duties  on  imports  and  interest  on  the 
public  debt. 

Since  Jan.  1,  1879,  they  have  been  redeemable  in  coin  at  the  office  of  the 
Assistant  Treasurers  of  the  United  States  in  the  Cities  of  New  York  and  San 
Francisco,  in  sums  of  not  less  than  $50.  They  represent  the  values  of  $1,  $2, 
$5,  $10,  $20,  $50,  $100,  $500,  $1000,  $5000,  $10,000.  The  Act  of  May  31, 
1878,  fixed  their  value  at  $346,681,016,  and  forbade  their  further  contraction. 

*  "  Any  gold  coin  of  the  United  States,  if  reduced  in  weight  by  natural  abrasion  not 
more  than  one-half  of  one  per  centum  below  the  standard  weight  prescribed  by  law,  after  a 
circulation  of  twenty  years,  as  shown  by  its  date  of  coinage,  and  at  a  ratable  proportion  for 
any  period  less  than  twenty  years,  is  received  at  its  nominal  vaiue  by  the  United  States 
treasury  and  its  offices."  The  "  Coinage  Act  of  1873  "  allows  a  deviation  from  the  standard 
weight  of  {  of  a  grain,  or  less,  in  the  manufacture  of  the  dollar  piece. 


Art.  120.]  UNITED     STATES     MONET.  43 

120.  National  Bank  Notes  (64O)  are  not  a  legal  tender  ; 
but,  since  they  are  secured  by  bonds  of  the  United  States  deposited 
with  the  U.  S.  Treasurer  at  Washington,  and  are  redeemed  in 
lawful  money   by  the  national  banks  and  the  Treasurer  of  the 
United  States,  they  are  usually  accepted  in  the  payment  of  debts 
in  any  part  of  the  United  States. 

They  are  receivable  in  all  parts  of  the  United  States  in  payment  of  taxes 
and  other  dues  to  the  United  States  except  duties  on  imports,  and  for  debts 
owing  by  the  United  States  to  individuals  and  corporations,  within  the 
United  States  except  interest  on  the  public  debt. 

They  represent  the  values  of  $1,  $2,  $5,  $10,  $20,  $50,  $100,  $500,  and  $1000. 
Since  Jan.  1, 1879,  no  notes  of  the  denomination  of  $1  and  $2  have  been  issued 
to  national  banks.  Nov.  1,  1886,  their  total  circulation  was  $301,529,889. 

121.  To  write  United  States  money. 

122.  In  writing  U.  S.  money,  the  decimal  notation  is  used. 
Dollars  are  written  at  the  left  of  the  decimal  point  and  form  the 
integral  part.     Cents  are  written  as  hundredths  of  a  dollar,  and 
occupy   the   first  two  places  at  the  right  of  the  decimal  point. 
Mills  are  written  as  thousandths  of  a  dollar,  and  occupy  the  third 
decimal  place. 

Thus,  twelve  dollars,  forty-eight  cents,  and  six  mills,  is  written  $12.486. 

When  the  number  of  cents  is  less  than  ten,  a  cipher  must  be  written  in 
the  first  place  at  the  right  of  the  point.  Thus,  eight  dollars  and  six  cents  is 
written,  $8.06. 

In  the  final  results  of  business  operations,  if  the  mills  are  more  than  five, 
they  are  regarded  as  an  additional  cent  ;  if  less  than  five,  they  are  rejected. 

In  checks,  notes,  drafts,  etc.,  cents  are  usually  written  as  hundredths  of  a 
dollar  in  the  form  of  a  fraction.  Thus,  twenty-five  cents  may  be  written,$T2^. 

123.  Express  the  following  amounts  by  figures  : 

1.  Eighty-six  dollars,  nineteen  cents,  five  mills. 

2.  Fourteen  dollars,  seventy-five  cents,  three  mills. 

3.  Five  hundred  twenty-six  dollars,  seventy  cents. 

4.  Two  thousand  dollars,  thirty  cents,  two  mills. 

5.  Seven  hundred  dollars,  nine  cents. 

6.  Fifty  thousand  dollars,  seven  mills. 

•7.  Four  hundred  eight  dollars,  two  cents,  five  mills. 
8.  Two  hundred  fifty  dollars,  sixty  cents,  three  mills. 


44  UNITED     STATES     MONEY.  [Art.  124. 

124.  To  reduce    dollars    to    cents  and  mills,  or  to 
reduce  cents  and  mills  to  dollars. 

125.  Dollars  may  be  reduced  to  cents  by  multiplying  by  100 
or  by  annexing  two  ciphers.     Dollars  may  be  reduced  to  mills  by 
multiplying  by  1000  or  by  annexing  three  ciphers. 

Thus,  $64  =  6400  cents,  or  64000  mills. 

If  the  amount  consists  of  dollars  and  cents,  reduce  to  cents  by  removing 
the  decimal  point  2  places  to  the  right ;  to  mills,  three  places  to  the  right. 
Thus,  $17.28  =  1728  cents,  or  17280  mills  ;  $34.658  =  34658  mills. 

126.  Cents  may  be  reduced  to  dollars  by  dividing  by  100,  or 
by  pointing  off  two  decimal  places.     Mills   may   be   reduced   to 
dollars  by  dividing  by  1000,    or  by  pointing  off  three   decimal 
places. 

Thus,  12345  cents  =  $123.45  ;  37560  mills  =  $37.56. 

127.  Reduce  the  following  to  cents  : 

1.  $345.  4.     $17.04.  7.     $148.19 

2.  $2376.  5.     $28.37.  8.     $204.40. 
8.     $2004.                  6.     $49.75.  9.     $317.04. 

128.  Change  the  following  to  mills  : 

1.  75  cents.  4.     $38.  '  7.  $14.172. 

2.  19  cents.              5.     $376.  8.  $4.866. 
8.     47  cents.               6.     $408.  9.  $10.012. 

129.  Reduce  the  following  to  dollars  : 

1.  148  cents.  4.     705  cents.  7.     18000  mills. 

2.  2300  cents.  5.     4212  cents.  8.     9370  mills. 
8.     4617  cents.           6.     13409  cents.  9.     12375  mills. 

130.  To  add  or  subtract  in  United  States  money. 

131.  RULE. —  Write  dollars  under    dollars,    and    cents 
under  cents.     Add  or  subtract  as  in  simple  numbers,  and 
place  the  point  in  the  result  directly  under  the  points  in 
the  numbers  added  or  subtracted. 

NOTE. — In  subtraction  of  U.  S.  money,  if  there  are  cents  in  the  subtrahend 
and  none  in  the  minuend,  suppose  ciphers  to  be  added  to  the  subtrahend  in 
cents'  place. 


Art.  132.]  UNITED     STATES     MONEY.  45 


EXAM  PLES. 

132.  1.  Add  5  dollars,   16  cents ;  18  dollars,  5  cents ;  404 
dollars,  75  cents ;  25  dollars,  8  cents ;  2376  dollars,  40  cents ;  8 
dollars,  2  cents. 

2.  Add  $170,  $106.40,  $240,  $200.40,  $70,  $.70,  $234.75.  ' 

3.  Add  $108.25,  $2345,  $6.04,  $7.10,  $192.43,  $117.05. 

4.  Add   $.06,    $6,    $108.16,   $500.64,    $564,    $5.64,    $117.10, 
$2081.48. 

5.  From  $124.16  subtract  $109.25. 

6.  From  $117  subtract  $98.49. 

7.  From  $575  subtract  575  cents. 

8.  A   merchant   makes  the   following    deposits   in   a  bank  i 
$1875.24,  $416,  $234.70,   $558.96,   and  $437.10.     He  draws  the 
following    checks:    $442.37,    $120.92,    $316.75,    $242.71,    $195, 
$716.32,  $100.48,  and  $76.19.     What  is  the  balance  of  his  bank 
account  ? 

133.  To  multiply  United  States  money. 

Ex.    Find  the  cost  of  9  desks  at  $2.45  each. 

OPERATION. 

2  45  ANALYSIS. — Disregarding  the  decimal  point,  multiply  as  in 

q  ordinary  multiplication.     9  times  245  cents  is  2205  cents  = 

$22.05. 
22.05 

134.  RULE. — Multiply  as  in  simple  numbers,  and  from 
the  right  of  the  product  point  off  as  many  figures  as  there 
are   figures    to    the  right  of  the  decimal  points  in  both 
numbers. 

NOTE. — If,  as  in  Examples  6  and  7,  the  number  expressing  cents  would 
make  a  convenient  multiplier,  use  it  as  such,  considered  as  an  abstract 
number,  and  point  off  the  result  according  to  the  rule. 

135.  Multiply  Multiply 

1.  12  dollars  and  18  cents  by  8.  7.  $.07  by  1239  ;  by  13416. 

2.  4  dollars  and  25  cents  by  12.  8.  $20.04  by  20  ;  by  108. 

3.  16  dollars  and  9  cents  by  17.  9.  $176  by  18  ;  by  144. 

4.  27  dollars  and  8  cents  by  25.  10.  $36.25  by  36  ;  by  117. 

5.  43  dollars  and  50  cents  by  76.  11.  $48.19  by  48  ;  by  288. 

6.  8  cents  by  2345  ;  by  3456.  12.  $50.08  by  75  5  by  192. 


40  UNITED     STATES     MONEY.  [Art.  136. 

136.  To  divide  in  United  States  money. 

Ex.    If  9  desks  are  worth  $22. 05,  what  is  one  worth  ? 

OPERATION.  ANALYSIS.— If  9  desks  are  worth  2205  cents  ($22.05),  1 

9  )  22.05  desk  is  worth  one-ninth  of  2205  cents  or  245  cents.     245  cents 

$2.45          -  $2.45. 

Ex.    If  8  chairs  are  worth  $18,  what  is  one  worth  ? 

OPERATION.  ANALYSIS. — If  the  dividend  consists  of  dollars  only,  and 

8  )  18.00  does  not  contain  the  divisor  an  exact  number  of  times,  reduce 

*2  ~T  it  to  cents  by  annexing  two  ciphers. 

Ex.    At  $6.25  each,  how  many  sheep  can  be  bought  for  $50  ? 

OPERATION.  ANALYSIS. — If    1    sheep    costs    $6.25,   as  many 

$6.25  )  $50.00  (  8         sheep  can  be  bought  for  $50  as  $6.25  is  contained 

times  in  $50.    $50  =  5000  cents.    $6.25  =  625  cents. 

Or>  5000  cents  -f-  625  cents  =  8  times.     Hence  the  result 

625.  )  5000.  (8  is  8  sheep. 

137.  RULE. — Divide  as  in  simple,  numbers,  and  point  off 
from  the  right  of  the  quotient  as  many  decimal  places  as 
those  in  the  dividend  exceed  those  in  the  divisor. 

NOTE. — If  the  divisor  alone  contains  cents,  make  the  dividend  cents  by 
annexing  two  ciphers  ;  or,  reduce  both  divisor  and  dividend  to  cents  by 
annexing  ciphers,  omit  the  decimal  points,  and  divide  as  in  simple  numbers. 

EXAM  PLES. 

138.  1.  If  12  books  are  sold  for  $41.40,  what  is  the  price  of 
one  book  ? 

2.  How  many  pounds  of  tea  at  65  cents  per  pound  can  be 
bought  for  $9.75? 

NOTE. — In  the  following  examples,  if  the  quotient  is  in  U.  S.  money  and 
the  result  is  not  an  exact  number  of  dollars,  continue  the  division  to  cents. 

Divide  Divide 

3.  $25.44  by  48  ;  by  106.  9.  $130.38  by  $2.46  ;  by  $1.06. 

4.  $476  by  25  ;  by  35.  10.  $149.04  by  $0.36  ;  by  $2.07. 

5.  $1728  by  36  ;  by  48.  11.  $156.24  by  $0.72  ;  by  $4.34. 

6.  $73.08  by  84  ;  by  87.  12.  $1728  by  $0.75  ;  by  $6.75. 

7.  $106.56  by  72  ;  by  576.  13.  $3456  by  $2.25  ;  by  $13.50. 

8.  $1884  by  75  ;  by  1535.  14.  $7154  by  $1.75  ;  by  $25.55. 


Art.  139.]  REVIEW     EXAMPLES.  47 

REVIEW     EXAMPLES. 

139.  1.  Find  the  sum  of  the  following  numbers  :  Twenty- 
six  thousand  forty-eight ;  twelve  thousand  four  hundred  eighty  ; 
one  hundred  thirty-six  thousand  ;  seven  hundred  ninety  thousand 
forty-three  ;  four  million  fifty-eight. 

2.  Subtract  eight  hundred  fourteen  thousand  nine  hundred 
sixteen  from  four  million  nineteen  thousand. 

3.  Multiply  five  hundred  sixty  thousand  seven  hundred  eight 
by  eighteen  hundred  sixty. 

4.  A  quantity  of  merchandise  was  bought  for  $27618.75,  and 
sold  for  $32418.25.     What  was  the  gain  ? 

5.  Find  the  total  length  of  the  Brooklyn  bridge,    its  meas- 
urements being  as  follows  :    Length  of  river  span,  1596  feet ;  of' 
each  (2)  land  span,  930  feet ;  of  New  York  approach,  1562  feet ; 
of  Brooklyn  approach,  971  feet. 

6.  If  I  sell  goods  for  $23876,  and  gain  $5389,  what  did  the 
goods  cost  me  ? 

7.  The  estimated  production  of  gold  and  silver  of  the  world 
for  1884  was  as  follows  :  Gold,  $98,990,772  ;  silver,  $116,525,949. 
For  1885,  gold,   $101,562,748;  silver,  $124,968,784.     What  was 
the  total  increase  ? 

8.  If  the  quotient  is  375  and  the  divisor  246,  what  is  the 
dividend  ? 

9.  If  the  product  of  two  numbers  is  450072,  and  one  of  the 
numbers  is  987,  what  is  the  other  number  ? 

10.  Divide   76432801   by  783.     Prove   that   your   solution   is 
correct. 

11.  A  clerk  receiving  a  salary  of  $1256,  pays  $468  a  year  for 
board,    $180   for  clothing,  and  $150  for  other  expenses.     What 
amount  has  he  left  ? 

12.  If  I  take  24889  from  the  sum  of  9872  and  24967,  divide 
the  remainder  by  50,  and  multiply  the  quotient  by  18,  what  is  the 
product  ? 

IS.  If  160  acres  of  land  cost  $10720,  how  many  acres  can  be 
bought  for  $8844  ? 

14.  If  75  head  of  cattle  cost  $2550,  what  will  59  head  cost  ? 

15.  Cash  on  hand  at  beginning  of  the  day,  $6492.75;  cash 
received,  $11456.75;   cash  paid  out,  $13285.26.      Kequired  the 
cash  balance  at  the  end  of  the  day. 


48  UNITED     STATES    MONEY.  [Art.  139. 

16.  A  merchant  sold  426  barrels  of  flour  for  $2556,  which  was 
$639    more    than   he    gave   for   it.      What   did   it   cost   him   a 
barrel  ? 

17.  Mr.  A  has  three  farms,  the  first  of  which  contains  158 
acres,  the  second  32  acres  less  than  the  first,  and  the  third  as 
many   as   the  other  two.     What  is  the  value  per  acre,  if  all  are 
worth  $26128  ? 

18.  A  merchant  bought  387  yards  of  cloth  at  79  cts.  per  yard  ; 
he  sold  298  yards  at  $1.16  per  yard,  and  the  remainder  at  97  cts. 
per  yard  ;  how  much  did  he  gain  ? 

19.  The  United  States  nickel  and  copper  coinage  for  the  year 
1886   was   5,519   five-cent   pieces,   4,519   three-cent   pieces,   and 
1,696,613  one-cent  pieces.     Find  total  value  of  minor  coinage. 

20.  The  silver  coinage  for  1886  was  as  follows:   29,838,905 
dollars,  6,105  half-dollars,  14,505  quarter-dollars,  1,767, 642  dimes. 
What  was  the  total  value  of  the  silver  coinage  ? 

21.  The  gold  coinage  for  1886  was  as  follows  :  243,584  double- 
eagles,    1,042,847   eagles,  3,751,629  half-eagles,  101  three-dollar 
pieces,  4,086  quarter-eagles,  8,567  one-dollar  pieces.     What  was 
the  value  of  the  gold  coinage  ? 

22.  There  are  four  bidders  to  supply  the  government  with  800 
tons  Lehigh,  500  tons  Cumberland,  and  700  tons  Baltimore  coal. 
A  offers  Lehigh  at  $6.29,  Cumberland  at  $4.38,  and  Baltimore  at 
$7.23.     B  offers  Lehigh  at  $6.80,  Cumberland  at  $4.12,  and  Balti- 
more at  $7.24.     C  offers  Lehigh  at  $6.40,  Cumberland  at  $4.45, 
and  Baltimore  at  $7.18.     D  offers  Lehigh  at  $6.17,  Cumberland  at 
$4.19,    Baltimore  at  $7.20.     Who  is  the  lowest  bidder  for  the 
whole  amount,  and  how  much  does  each  bid  amount  to  ? 

23.  A  drover  bought  a  number  of  cattle  for  $12204,  and  sold 
the  same  for  $13560,  by  which  he  gained  $4  per  head.     How  many 
cattle  were  purchased  ? 

24»  A  farmer  raised  in  one  year  512  bushels  of  wheat,  the  next 
year  twice  as  much  as  he  raised  the  first  year,  and  the  third  year 
four  times  as  much  as  he  did  the  second  year.  What  was  the 
value  of  the  three  crops  at  $1.65  per  bushel  ? 

25.  Bought  75  tons  of  hay  at  $16  per  ton  ;  gave  in  payment 
56  sheep  at  $3.75  each,  and  the .  remainder  I  paid  in   butter  at 
33  cts.  per  pound.     How  many  pounds  of  butter  were  required  ? 

26.  Bought  225  acres  of  land  for  $12,600,  and  sold  116  acres  at 
$65  per  acre,  and  the  remainder  at  cost ;  how  much  did  I  gain  ? 


Art.  139.] 


PROPERTIES     OF     NUMBERS. 


49 


27.  A  sold  to  B  175  acres  of  land  at  $135  an  acre,  and  by  so 
doing  gained  $1925  ;  B  sold  the  land  at  a  loss  of  $1750.     What 
did  A  pay  per  acre,  and  what  was  B's  selling-price  per  acre  ? 

28.  A  merchant  sold  800  barrels  of  flour  for  $5867, 144  barrels 
of  which  he  sold  at  $7  per  barrel,  and  225  barrels  at  $6.75.     At 
how  much  per  barrel  did  he  sell  the  remainder  ? 

29.  According  to  the  following  table,  what  was  the  average 
immigration  per  year  ?     What  per  month  ? 


Years. 

Number. 

Years. 

Number. 

Years. 

Number. 

1875 

227  498 

1879 

177  826 

1883 

603  322 

1876  

169,986 

1880  

457  257 

1884 

518  592 

1877  

141,857 

1881   .  .  . 

669  431 

1885 

395  346 

1878  

138,469 

1882     . 

788  992 

1886 

334  203 

PROPERTIES     OF     NUMBERS. 

140.  A  Number  is  a  unit,  or  a  collection  of  units  ;  as  one, 
four,  three  feet,  five  dollars. 

141.  All  numbers  are  either  integral  or  fractional,  abstract 
or  concrete. 

142.  An  Integral  Number,  or  Integer  is  a  number  which 
expresses  whole  things  ;  as  two,  four  gallons,  seven  dollars. 

143.  A  Fractional   Number,  or  Fraction   is  a  number 
which  expresses  one  or  more  equal  parts  of  a  unit ;  as  one-half, 
three-fourths. 

144.  An  Abstract  Number  is  a  number  which  does  not 
refer  to  any  particular  object ;  as  one,  six,  ten. 

145.  A  Concrete  Number  is  a  number  applied  to  an  object, 
or  quantity ;  as  three  apples,  five  pounds,  ten  dollars. 

146.  Integral   numbers    are   either   odd    or    even,   prime  or 
composite. 

147.  An  Odd   Number  is  a  number   whose  unit  figure  is 
1,  3,  5,  7,  or  9  ;  as  7,  21,  39. 


50  PROPERTIES     OF    NUMBERS.  [Art.  148. 

14:8.  An  Even  Number  is  a  number  whose  unit  figure  is 
0,  2,  4,  6,  or  8  ;  as  6,  40,  74. 

149.  A  Prime  Number  is  a  number  which  can  be  exactly 
divided  only  by  itself  and  unity ;  as  1,  1,  13,  29. 

150.  Numbers  are  prime  to  each  other  when  no  integral 
number  greater  than  1  will  divide  each  without  a  remainder. 

Numbers  that  are  prime  to  each  other  are  not  necessarily  prime  numbers. 
Thus,  25  and  28  are  prime  to  each  other,  but  they  are  not  prime  numbers. 

151.  A  Composite   Number  is   a  number  which   can  l;e 
exactly  divided  by  other  integers  besides  itself  and  unity. 

Thus  28,  the  product  of  4  and  7,  is  a  composite  number.     It  is  exactly 
divisible  by  4  and  7. 


DIVISIBILITY    OF    NUMBERS. 

152.  An  Exact   Divisor  of  a  number  is  any  number  that 
will  divide  it  without  a  remainder. 

Thus  2,  3,  4,  6,  8,  and  12  are  exact  divisors  of  24. 

153.  A  number  is  said  to  be  divisible  by  another  when  the 
latter  will  divide  the  former  without  a  remainder.     Any  number 
is  divisible 

1.  By  2,  if  it  is  an  even  number  ;  as  6,  28,  and  32. 

2.  By  3,  if  the  sum  of  its  digits  is  divisible   by  3  ;  as   841) 
(8  +  4  +  9  =  21,  21  is  divisible  by  3),  7323,  and  47892. 

3.  By  4,  if  the  two  right-hand  figures  are  ciphers,  or  express  a 
number  divisible  by  4  ;  as  1100,  216,  and  7328. 

4.  By  5,  if  the  right-hand  figure  is  0  or  5  ;  as  40  and  135. 

5.  By  6,  if  it  is  an  even  number  and  the  sum  of  its  digits  is 
divisible  by  3  ;  as  216,  840,  and  732. 

6.  By  8,  if  the  three  right-hand  figures  are  ciphers,  or  express 
a  number  divisible  by  8  ;  as  3000  and  7168. 

7.  By  9,  if  the  sum  of  its  digits  is  divisible  by  9 ;  as  216,  783, 
and  12348. 


Art.  154.]  PRIME     FACTORS.  51 


PRIME     FACTORS. 

154.  The  Factors  of  a  number  are  those  numbers  which 
when  multiplied  together  will  produce  the  number. 

Thus  4  and  7;  2  and  14;  2,  2,  and  7  are  factors  of  28.    The  number  itself 
and  unity  are  not  regarded  as  factors. 

The  factors  of  a  number  are  also  the  exact  divisors  of  it. 

155.  A  Prime  Factor  is  a  prime  number  used  as  a  factor. 

Thus,  2,  2,  and  7  are  the  prime  factors  of  28.     4  is  a  factor  of  28,  but  not 
&  prime  factor. 

156.  To   find  all  the  prime  factors  of  a  composite 
number. 

Ex.    What  are  the  prime  factors  of  6930. 

OPERATION.  ANALYSIS. — Any  prime  number  that  is  an  exact  divi- 

2  )  6930  sor  of  the  given  number  is  a  prime  factor  of  it.     Divide 
o  \  QAAP;           ^e  given  number  by  2  (153,  1),  the  least  prime  divisor  of 

it,  obtaining  the   quotient  3465.    Next,  divide  this  quo- 

3  )_1155  tient  successively  by  3  (153,  2),  3,  5  (153,  4),  and  7. 
5  }  385  ^ke  *ast  quotient;  11  ig  a  Prirae  number  and  therefore  a 

prime  factor.     The  several   divisors  2,  3,  3,  5,  7  and  the 
7  )  7?  last  quotient  11  are  the  prime  factors  required. 

11  2x3x3x5x7x11  = 


157.  RULE. — Divide  by  the  least  prime  number  which 
will   divide  the  given  number  without  a  remainder.    In 
like   manner  divide  the  resulting  quotient,  and  continue 
the  division  until  the  quotient  is  a  prime  number.     Tl^e 
several  divisors  and  the  last  quotient  are  the  prime  factors. 

EXAMPLES. 

158.  Resolve  the  following  numbers  into  their  prime  factors  : 

1.  3465.  r.     6552.  13.     8192.  19.     6660. 

2.  3003. 

3.  4158. 

4.  3150. 

5.  3675. 

6.  2310. 


8.  7826. 

14.  6561. 

20.  2448. 

9.  6006. 

15.  3125. 

21.  8525. 

10.  5368. 

16.  1800. 

22.  9926. 

11.  3825. 

17.  1935. 

23.  9576. 

12.  5324. 

18.  2475. 

24.  5075. 

52  PROPERTIES     OF    NUMBERS.  [Art.  159. 


COMMON     MULTIPLES.* 

159.  A  Multiple  of  a  number  is  a  number  that  is  exactly 
divisible  by  it  ;  or,  it  is  any  product  of  which  the  given  number 
is  a  factor. 

Thus,  12  is  a  multiple  of  6;  15  of  5;  etc. 

160.  A  Common  Multiple  of   two  or  more  numbers  is  a 
number  that  is  exactly  divisible  by  each  of  them. 

Thus,  12,  24,  36,  and  48  are  multiples  of  4  and  6. 

161.  The  Least  Common  Multiple  of  two  or  more  num- 
bers is  the  least  number  that  is  exactly  divisible  by  each  of  them. 

Thus,  12  is  the  least  common  multiple  of  4  and  6. 


2.    PRINCIPLES.  —  1.  A  multiple  of  a  number  contains  all 
the  prime  factors  of  that  number. 

2.  A  common  multiple  of  two  or  more  numbers  contains  all  the 
prime  factors  of  each  of  those  numbers. 

3.  The  least  common  multiple  of  two  or  more  numbers  contains 
all  the  prime  factors   of  each   of  the   numbers,    and    no    other 
factors. 

163.   To   find  the   least  common  multiple  of  two  or 
more  numbers. 

Ex.     What   is   the    least    common   multiple   of    12,    18,    20, 
and  40  ? 

FIRST  OPERATION.  ANALYSIS.  —  Since  40,    a  multiple 

12  =  2x2x3  of  20,  contains  all  the  prime  factors  of 

18  =  2x3x3.  20,  the  number  20  may  be  omitted  in 

40  =  2x2x2x5  ^e  °Perati°n-     Resolve  the  numbers 

into  their  prime  factors.     The  least 

2x2x2x3x3x5  =  360         common  multiple  must  contain  2  as  a 

factor  3  times  in  order  to  be  divisible 

by  40  ;  it  must  contain  8  as  a  factor  twice  in  order  to  be  divisible  by  18  ;  and 
it  must  contain  5  as  a  factor,  in  order  to  be  divisible  by  40.  360,  the  product 
of  the  factors,  2,  2,  2,  3,  3,  and  5,  is  the  least  common  multiple  of  the  given 
numbers,  since  it  contains  the  different  factors  the  greatest  number  of  times 
that  they  occur  in  the  given  numbers,  and  no  other  factors  (Prin.  3). 

*  For  Greatest  Common  Divisor,  see  Appendix,  page  317. 


Art.  1 63.J  C  0  MM  ON    MUL  TIP  LES.  53 

SECOND  OPERATION.  ANALYSIS.— The      factors    of     the      F6- 

2  )  12,      18,      40  quired  multiple  may   be  found  by   the 

%  \  Q g on  following  process.   Divide  the  given  num- 

bers by  any  prime  number  that  will  divide 

3)3,        9,      10  two  or  more  of  them,  writing  the  quo- 

-i          o       10  tients  and  the  undivided  numbers    be- 

neath.    Treat  the  resulting  numbers  in 

2x2x3x3x10  =  360         like   manner,    and  continue  the  process 

until  no  two  of  the  numbers  have  a  com- 
mon factor  or  divisor.  The  product  of  the  several  divisors  and  the  remaining 
quotients  and  undivided  numbers  will  be  the  least  common  multiple. 

164.  BULE. — Resolve  the  given  numbers  into  their  prime 
factors.     The  product  of  the  different  prime  factors,  taking 
each  factor  the  greatest  number  of  tunes  it  appears  in  any 
of  the  numbers,  will  be  the  least  common  multiple.     Or, 

Divide  the  given  numbers  by  any  prime  number  (see 
Note  2)  that  will  exactly  divide  two  or  more  of  them,  writing 
the  quotients  and  undivided  numbers  beneath.  Repeat  the 
operation  with  the  resulting  numbers  until  there  is  no  exact 
divisor  of  any  two  of  them.  Tlie  product  of  the  several 
divisors  and  the  last  quotients  and  undivided  numbers  will 
be  the  least  common  multiple. 

NOTES. — 1.  In  the  operation,  reject  such  of  the  smaller  numbers  as  are 
divisors  of  the  larger ;  also  reject  such  of  the  quotients  and  undivided  num- 
bers as  are  divisors  of  the  others. 

2.  Divide  by  composite  numbers  when  they  are  exact  divisors  of  all  the 
numbers. 

EXAMPLES. 

165.  Find  the  least  common  multiple  of  the  following  numbers: 

1.  6,  10,  15,  and  30.  12.     24,  36,  and  40. 

2.  16,  24,  and  48.  13.     32,  48,  and  72. 

8.  30,  40,  and  60.  U.  16,  22,  24,  and  30. 

4.  2,  4,  8,  and  16.  15.  18,  28,  30,  and  36. 

6.  14,  21,  and  28.  16.  12,  16,  20,  and  24. 

6.  5,  8,  15,  and  18.  17.  33,  44,  55,  and  66. 

7.  6,  9,  21,  and  24.  18.  27,  36,  42,  and  48. 

8.  12,  20,  and  30.  19.  36,  45,  60,  and  72. 

9.  6,  10,  30,  and  40.  20.  28,  35,  42,  and  56. 

10.  32,  48,  and  60.         21.  45,  55,  60,  and  75. 

11.  24,  32,  and  40.         22.  60,  72,  84,  and  90, 


54  PROPERTIES     OF    NUMBERS.  [Art.  166. 


CANCELLATION. 

166*  Cancellation  is  a  method  of  shortening  an  operation 
by  rejecting  equal  factors  from  both  dividend  and  divisor. 

167.  PRINCIPLE. — Dividing  both  dividend  and  divisor  by  the 
same  number  does  not  affect  the  value  of  the  quotient. 

Ex.    Divide  84  x  36  by  27  x  14. 

OPERATIONS.  ANALYSIS. — Indicate  the  oper- 

2  Or,  ations  to  be  performed  as  in  the 

*  2          margin.    Since  36  and  27  contain 

=  3  */l       **  the  common  factor  9,  cancel  or  re- 

10  4  ject  it  from  both,  retaining  the 

factors  4  and  3  respectively.  14 
and  84  contain  the  common  factor 
14  ;  therefore  reject  it,  retaining 

the  factor  6  in  the  dividend.  [Since  cancellation  is  a  process  of  division,  the 
rejecting  of  14  does  not  destroy  it,  but  divides  it,  leaving  1  as  a  quotient.  It 
is  unnecessary  to  write  1  as  a  quotient,  except  when  there  are  no  other  factors 
in  the  dividend.]  3  is  a  common  factor  of  6  and  3  ;  therefore  reject  it  from 
both,  retaining  the  factor  2  in  the  dividend.  The  product  of  the  remaining 
factors,  2  and  4,  is  the  required  quotient. 

168.  RULE. — Write  the  numbers  denoting  multiplication 
above  a  horizontal  Hue,  and  the  numbers  denoting  division 
below.    The  numbers  above  the  line  will  form  a  dividend,  and 
the  numbers  below,  a  divisor.     Cancel  the  factors  common 
to  both  dividend  and  divisor.    Tlie  product  of  the  remaining 
factors  of  the  dividend  divided  by  the  product  of  the  re- 
maining factors  of  the  divisor  will  be  the  required  quotient. 

EXAMPLES. 

169.  Find  the  value  of  the  following  expressions  : 

27  x  48  x  60                1760x175x6  360  x  28  x  27  x  5 

*  54x36^40*         *"    36  x  100  x  10  '  *  25  x  42  x  18x~12' 

1500x144x5            40  x  36  x  42  x  18  17x36x25x144 

'       365x100             '     9x35x30x8  *  '  48x60x106x51' 

1760  x  6  x  145            24  x  30  x  54  x  35  144  x  625  x  37  x  12 

100x365             '  14x15x21x64*  288x^75x185 

10.  Multiply  72  by  3  x  18,  divide  the  product  by  8  times  9, 
multiply  the  quotient  by  7  x  20,  divide  the  product  by  360,  mul- 
tiply the  quotient  by  6  times  8. 


Art.  169.]  CANCELLATION.  55 

11.  If  42  tons  of  coal  cost  $147,  what  will  16  tons  cost  ? 

12.  A  man  gave  9  pounds  of  butter  at  17  cents  a  pound  for 
3   gallons  of  molasses ;    how  much  was  the  molasses  worth  a 
gallon  ? 

IS.  If  20  pounds  of  beef  cost  250  cents,  what  cost  75  pounds  ? 

14.  How  many  potatoes  at  65  cents  per  bushel  will  pay  for 
13  weeks'  board  at  $7.50  per  week  ? 

15.  A  merchant  bought  375  barrels  of  flour  at  $5.50  per  barrel, 
and  paid  in  cloth  at  $2.75  per  yard ;    how  many  yards  did  it 
require  ? 

16.  How  many  pounds  of  coffee  at  27  cents  per  pound  should 
be  given  for  57  bushels  of  corn  at  63  cents  per  bushel  ? 

17.  Sold  28  bushels  of  apples  for  $21 ;  what  should  I  receive 
for  42  bushels  ? 

18.  How  many  cows  worth  $35  each  must  be  given  in  exchange 
for  84  tons  of  hay  at  $15  per  ton  ? 

19.  How  many  bushels  of  corn  at  52  cents  a  bushel  must  be 
exchanged  for  324  bushels  of  oats  at  39  cents  per  bushel  ? 

20.  If  430  bushels  of  wheat  are  obtained  from  sowing  7  bush- 
els, how  much  would  be  obtained  from  sowing  21  bushels  ? 

21.  What  should  be  paid  for  the  transportation  of  3600  pounds 
of  cheese  at  the  rate  of  47  cents  per  100  pounds  ? 

22.  What  must  be  paid  for  transporting  31600  pounds  of  iron 
at  $5  per  ton  of  2000  pounds  ? 

23.  What  will  7840  pounds  of  coal  cost,  at  $6  per  ton  of 
2240  pounds  ? 

24.  If  3  men  eat  7  pounds  of  meat  in  one  week,  how  much 
would  6  men  eat  in  4  weeks  ? 

25.  How  many  canisters,  each  holding  40  ounces,  can  be  filled 
from  3  chests  of  tea,  each  containing  55  pounds  of  16  ounces  ? 

26.  How  many  times  can  16  bottles,  each  holding  3  pints,  be 
filled  from  6  demijohns,  each  containing  10  gallons  of  8  pints 
each  ? 

27.  A  man  exchanged  275  barrels  of  potatoes,  each  containing 
3  bushels,  at  54  cents  per  bushel,  for  a  certain  number  of  pieces 
of  muslin  each  containing  45  yards,  at  11  cents  per  yard.     How 
many  yards  did  he  receive  ? 

28.  If  a  person  travel  24  hours  each  day  at  the  rate  of  45  miles 
an  hour,  how  many  days  would  it  require  to  pass  around  the 
globe,  a  distance  of  25000  miles  ? 


FRACTIONS. 


170.  A  Fraction  is  one  or  more  of  the  equal  parts  of  a  unit  ; 

as  one-half  (J),  two-thirds  (f  ),  one-fourth  (£),  seven-eights  (J). 

If  a  unit  be  divided  into  four  equal  parts,  each  part  is  called  a  fourth.  If 
one  of  these  parts  be  taken,  the  expression  will  be  one-fourth  (\)  ;  if  three 
parts,  three-fourths  (f),  etc. 

171.  The  greater  the  number  of  equal  parts  into  which  a  unit 
is  divided,  the  less  will  be  each  part  ;  the  less  the  number  of  parts, 
the  greater  will  be  each  part. 

One-half  (|)  is  greater  than  one-third  (£)  ;  one-fourth  (J)  is  less  than  one- 
third  i). 


A  fraction  is  usually  expressed  by  two  numbers,  one 
written  above  the  other,  with  a  line  between.  Fractions  written 
in  this  form  are  called  Common  Fractions. 

173.  The  number  below  the  line  is  called  the  Denominator, 
because  while  indicating  the  number  of  equal  parts  into  which 
the  unit  is  divided,  it  denominates  or  names  those  parts. 

174.  The  number  above  the  line  is  called  the  Numerator, 
because  it  shows  how  many  of  the  parts  are  taken  to  form  the 
fraction. 

175.  The  numerator   and  denominator,  taken  together,  are 
called  the  Terms  of  the  fraction. 

In  the  fraction  f  ,  3  and  4  are  the  terms  ;  4  is  the  denominator,  and  shows 
that  the  unit  is  divided  into  four  equal  parts,  called  fourths  ;  3  is  the  numera- 
tor, and  shows  that  three  of  these  parts  are  taken  to  constitute  the  fraction. 

176.  A  fraction   is   an   expression  of  unperformed   division 
The  numerator  is  the  dividend,  the  denominator  is  the  divisor, 
and  the  value  of  the  fraction  is  the  quotient. 

177.  A  Simple  Fraction  is  a  single  fraction,  both  of  whose 
terms  are  integers. 

178.  Simple  fractions  are  proper  or  improper. 


Art.  179.]  FRACTIONS.  57 

179.  A  Proper  Fraction  is  one  that  is  less  than  a  unit ;  the 
numerator  being  less  than  the  denominator.  Thus,  J-,  J,  and  | 
are  proper  fractions. 

ISO.  An  Improper  Fraction  is  one  that  is  equal  to,  or 
greater  than  a  unit ;  hence  the  numerator  must  be  equal  to,  or 
greater  than  the  denominator.  Thus,  f,  f,  f,  and  ty-  are  im- 
proper fractions. 

181.  A  Mixed  Number  is  an  integer  and  a  fraction  united  ; 
as  2J,  4f,  18}. 

182.  A  Complex  Fraction  is  one  whose  numerator  is  a 

f-     105f     75}     3|     12£     16| 
fraction  or  a  mixed  number;  as  |-,  -^,  —±,  -±-,  —±,  —  g. 

Q3 

The  expression  -j  indicates  division,    and    is   not   properly   a  fraction. 

^¥ 

A  unit  cannot  be  divided  into  5^  equal  parts. 

183.  PRINCIPLES. — 1.  Multiplying  the  numerator  or  dividing 
the  denominator  by  a  number  multiplies  the  fraction  by  that  number. 

2.  Dividing  the  numerator  or  multiplying  the  denominator  by  a 
number  divides  the  fraction  by  that  number. 

3.  Multiplying  or  dividing  both  numerator  and  denominator  by 
the  same  number  does  not  change  the  value  of  the  fraction. 

EXERCISES. 

184.  1.  Read  the  following  fractions,  and  copy  separately  : 
1,  the  simple  fractions  ;  2,  the  proper  fractions ;  3,  the  improper 
fractions  ;  4,  the  mixed  numbers  ;  5,  the  complex  fractions  : 

if;  H;V;   A-;  -J;  ¥;  tt;  ^l;  y;  *;  3i;  13S; 

\>     ft;     7*;     8*5     46|;     ifi;     || ;     ^ ;     f;     f;     f. 

#.  Write  the  following  fractions  :  three  fourths ;  seven  eighths ; 
nineteen  sixteenths  ;  five,  and  one  half ;  one  hundred  three  thirty- 
seconds  ;  one  hundred,  and  three  thirty-seconds  ;  forty-eight,  and 
five  twelfths ;  eleven  tenths  ;  nine  forty-fifths. 

3.  Write  the  following  fractions :  eight  ninths  ;  thirteen,  and 
two-thirds ;  sixteen  twenty-fourths ;  ten  tenths ;  fourteen,  and 
forty-six  hundredths  ;  sixty-five  forty-eighths  ;  nineteen  one  hun- 
dred nineteenths  ;  thirty-six  four  hundred  thirty-seconds. 


58  FRACTIONS.  [Art.  185. 


REDUCTION     OF    FRACTIONS. 

185.  Reduction  of  Fractions  is  the  changing  their  form 
without  changing  their  value. 

186.  A  fraction  is  reduced  to  lower  terms  when  the  numerator 
and  denominator  are  expressed  in  smaller  numbers. 

187.  A  fraction  is  in  its  lowest  terms  when  its  numerator  and 
denominator  have  no  common  divisor. 

188.  A  fraction  is  reduced  to  higher  terms  when  the  numerator 
and  denominator  are  expressed  in  larger  numbers. 

189.  To  reduce  a  fraction  to  its  lowest  terms. 

Ex.    Reduce  -ffa  to  its  lowest  terms. 

OPERATION.  ANALYSIS.  —  Dividing  both  terms  of  the  fraction, 

T/ir  —  iH"  —  f        T8A>  by  the  common  divisor,  6,   the   result   is  ff  ; 

dividing  both  terms  of  £f  by  the  common  divisor, 

7,  the  result  is  f.     Since  2  and  3  have  no  common  divisor,  the  fraction  is 
reduced  to  its  lowest  terms  (187). 

The  value  of  the  fraction  has  not  been  changed,  because  both  terms  have 
been  divided  by  the  same  number  (183,  3). 

190.  EULE.  —  Divide  the  terms  of  the  fraction  by  any 
number  that  ivill  divide  both  without   a  remainder,   and 
continue   the  operation  with  the  resulting  fractions  until 
they  have  no  common  divisor. 

EXAMPLES. 

191.  Reduce  to  their  lowest  terms, 

i-  if-         *•  i%.        17.  m-        &•  iff. 


a.  H-  10.  in-           18-    tti            **•    HI*. 

&  **•  11-  «*• 

4-  If.  12.  Hi 

5.  3%.  18.  iftt-  21.     Jfr.                 29. 

6.  -^  14.  f}f.  **.     H*.                 80. 
7-  1%.  15.  fWV-             28.     -MV.               81. 

8. 


Ait.  192.]  REDUCTION     OF    FRACTIONS.  59 

192.  To  reduce  a  fraction  to  higher  terms. 

Ex.    Reduce  f  to  a  fraction  whose  denominator  is  32. 

OPERATION.  ANALYSIS.— The  fraction  f  is  reduced  to  thirty- 

32  -r-  4  =  8         seconds,  without   changing   its  value,  by  multiplying 
|  '=  JJ-  the  terms  by  the  number  that  will  cause  its  denomina- 

tor 4  to  become  32  (183,  3).     By  dividing  the  required 

denominator  32  by  the  given  denominator  4,  this  number  is  found  to  be  8. 
Multiplying  both  terms  of  f  by  8,  the  result  is  ff .  In  practice,  say  or  think, 
4  into  32  8  times.  8  times  3  are  24. 

193.  RULE. — Divide  the  required  denominator  ~by  the 
denominator    of    the   given    fraction,    and    multiply   the 
numerator  of  the  given  fraction  by  the  quotient. 

EXAMPLES. 

194.  - 1.  Reduce  f  to  48ths. 

2.  Change  T7^  to    an   equivalent   fraction  having   60   for  its 
denominator. 

3.  Reduce  -|,  f,  £,  |  each  to  24lhs. 

4.  Reduce  £,  f,  T\,  J  each  to  36ths. 

5.  Reduce  |,  £,  TV  each  to  48ths. 

6.  Reduce  \-,  f,  f£    each  to  105ths. 

7.  Reduce  ffc,  f>  i  each  to  56ths- 

8.  Reduce  TV,  H>  if  eacn  to  96the. 
^.  Reduce  },  f,  -^  each  to  360ths. 

10.  Reduce  j|,  f,  ffc.    each  to  72ds. 
^.  Reduce  f,  f^,  |J  each  to  108ths. 
12.  Reduce  |,  |,  ^  each  to  360ths. 

195.  To  reduce  two  or  more  fractions  to  equivalent 
fractions  having  their  least  common  denominator. 

196.  A  Common  Denominator  of  two  or  more  fractions  is 
a  denominator  to  which  they  can  all  be  reduced,  and  is  the  com- 
mon multiple  of  their  denominators. 

197.  The  Least  Common  Denominator  of  two  or  more 
fractions  is  the  least  denominator  to  which  they  can  be  reduced, 
and  is  the  least  common  multiple  of  their  denominators. 


60 


FRACTIONS. 


[Art.  197. 


Ex.    Eeduce  f,  f  ,  $,  -fy  to  equivalent  fractions  having  their 
least  common  denominator. 


-|   —   44}- 

3    __   4  5. 

° 

S 

To  --    "So 


OPERATION. 
2  )   $,  4,   G, 


10 


2  x  2  X  3  X  5  =  60 


ANALYSIS.  —  The  least  common 
multiple  of  the  denominators  is 
found  to  be  60  (163),  which  we  take 
as  the  least  common  denominator. 
By  Art.  193,  |  is  reduced  to  «. 
We  proceed  in  the  same  manner 
with  each  of  the  other  fractions. 

The  value  of  each  fraction  remains  unchanged,  since  both  terms  have  been 
multiplied  by  the  same  number.  In  many  cases,  the  least  common  denomina- 
tor can  be  readily  found  by  inspection. 

198.  RULE.  —  Find  the  least  common  multiple  of  the 
given  denominators  for  the  least  common  denominator, 
and  reduce  the  given  fractions  to  this  denominator. 


EXAMPLES. 


199.  Reduce  the  following  fractions  to  equivalent  fractions 
having  their  least  common  denominator  : 

i-    iiVfV          5.    «,  «,  H-          9- 


.    3,  V,  A>  tt- 


,  t,  I- 


if  >  i- 


2OO.  To  reduce  an  integer  or  a  mixed  number  to  an 
improper  fraction. 

Ex.    In  18  units,  how  many  fourths  ? 

ANALYSIS.  —  In   1  there  are  4  fourths  (f),  and  in   18,  eighteen  times  4 
fourths,  or  72  fourths  (-\2-).     Hence,  18  =  *£. 

Ex.    Reduce  16  J  to  an  improper  fraction. 


OPEBATION. 

16} 

g 

- 
128  eighths. 

7  eighths. 
135  eighths. 


ANALYSIS.  —  In  1  there  are  8  eighths  (f  ),  and  in  16, 
sixteen  times  8  eighths,  or  128  eighths  (if  4).  128 
eighths  and  7  eighths  are  135  eighths.  Hence, 
16|  =  if*. 


Art.2Ol.J  REDUCTION     OF    FRACTIONS.  61 

201.  RULE.  —  Multiply  the  integer  by  the  required  denom- 
inator, and  to  the  product  add  the  numerator  of  the  frac- 
tion, and  under  the  result  write  the  denominator. 

NOTE.  —  When  the  numerator  of  the  fraction  is  a  small  number,  add  it 
mentally  to  the  product  of  the  integer  and  the  denominator. 

EXAM  PLES. 

202.  1.  In  27,  how  many  ninths  ? 

2.  Reduce  46  \  to  halves. 

3.  How  many  eighths  of  a  peck  in  37-J  pecks  ? 

Reduce  the  following  to  improper  fractions  : 

4.  37f  ;  19J  ;  208TV  9.     81|  ;  196J  ;  375f. 

5.  56}  ;  49£  ;  182f  10.     116ft  ;  456^  ;  87}}. 

6.  375*;  94^;  46}.  11.     24&  ;  179^}  ;  1767}. 

7.  44}  ;  37^  5  19«.  ^.     87}  ;  490^  ;  168ft. 

J^.  384f  ;  161}  ;  175f|. 


203.  To  reduce  an  improper  fraction  to  an  integer 
or  a  mixed  number. 

Ex.    Reduce  *£-  to  a  mixed  number. 

ANALYSIS.  —  1  =  |  ;  hence  in  -^-,  there  are  as  many  units  as  4  fourths  aro 
contained  times  in  27  fourths,  or  6|. 

204.  RULE.  —  Divide  the  numerator  ~by  the  denominator. 


EXAM  PLES. 

2O5.     1.  Change  -^  to  a  mixed  number. 
2.  Reduce  &£•  of  a  dollar  to  dollars. 
Reduce  to  integers  or  mixed  numbers  : 

S.      AJJL  ;   A^JL.  8.      Afjp-  ;   J-ff-*.  75.      ^  ; 

4.      l$a  ;   1|1.  P.       A}A  ;   ^L.  ^.       ^  ;   A}f i 

^.   H*  ;  A*A-         ^-   W ;  W-  ^   W ;  W- 

ft   W;W-          ^   W;W-  16-   W;1^ 

7.       ^  ;    J1JJ1.  ^.       ^L  ;    1||A.  17.       *ff.  ; 


FRACTIONS. 


ADDITION     OF     FRACTIONS. 

206.  Addition  of  Fractions  is  the  process  of  finding  the 
sum  of  two  or  more  fractions. 

207.  PRINCIPLE. — In  order  that  fractions  may  be  added,  they 
must  have  like  denominators  and  be  parts  of  like  units. 

Ex.   What  is  the  sum  of  -^  •&,  and  T^  ? 

OPERATTON.  ANALYSIS.— As    these    frac- 

&  ~f~  A  +  iV  —  If  —  f  —  li         ^ons  nave  a  common  denomina- 
tor, we  add  their  numerators, 

and  write  their  sum,  15,  over  the  common  denominator,  12.     |f  =  1|,  the 
required  result. 

Ex.    Add  f,  f,  and  |. 

OPERATION. 


ANALYSIS. — Reduce  the  given  fractions  to  equivalent  fractions  having  the 
least  common  denominator,  12  (198).     Then  proceed  as  in  previous  example. 

Ex.    Find  the  sum  of  29J-,  38},  17f,  and  42f 

OPERATION. 

24ths. 

4 

-i  o  ANALYSIS. — The  sum  of  the  fractions  is 

|f  =  If,  which  added  to  the  sum  of  the  inte- 
gers, gives  127|,  the  required  result. 


Ex.  How  many  yards  in  12  pieces  of  prints  containing  461, 
482,  512,  493,  441,  482,  471,  49,  473,  503,  481,  482  yards  respect- 
ively ? 

OPERATION. 

461  471 

482  49  ANALYSIS.— The  small  figures    represent 

512  4.73  fourths  (quarters).    The  sum  of  the  fourths  is 

493  503  "^  =  ^^'  wn^c^  ftdded  to  the  sum  of  the  in- 

A  .  l  .oi  tegers  gives  580£,  the  total  number  of  yards. 

482  482     5801. 


Art.  208.]  ADDITION     OF    FRACTIONS.  63 

208.  RULE. — Reduce  the  given  fractions  to  equivalent 
fractions  having  the  least  common  denominator.     Write 
the  sum  of  the  numerators  over  the  common  denominator, 
and  reduce  the  resulting  fraction  to  its  simplest  form. 

When  there  are  mixed  numbers  or  integers,  add  the 
integers  and  fractions  separately,  and  then  add  the  results. 

XOTE. — Before  adding,  reduce  all  fractions  to  their  lowest  terms,  and  all 
improper  fractions  to  mixed  numbers. 

i 
EXAMPLES. 

209.  Add  the  following  : 

-Z-  A,  H,  A,  and  if  6.  127&,  ft,  175},  and  f. 

2.  f,  },  },  and  }.  6.  141  A,  197},  and  43ft. 

S.  12},  7f,  16ft,  and  38}.  7.  75},  |,  1028},  and  }}. 

£  48|,  46ft,  31f,  and  17}.  S.  },  119},  240},  and  17ft. 

5.  461,  483,  402,  49,  473,  and  462.     (See  Analysis  opposite.) 

10.  403,  411,  482,  441,  493,  482,  493,  49^  473,  483,  483,  and  491. 

11.  18|,  27},  42},  51|,  and  14J-. 
J^.  146},  1^,  53ft,  and  68J. 

J5.     1172|,  19},  440J,  6},  and  10ft. 

14.  ft,  106ft,  37f,  7|,  and  176}}. 

15.  175,  116ft,  143},  and  £7f 

76.  20},  164,  tt>  and  43f  • 

77.  44},  16},  29ft,  13},  and  44}}. 
jtf.     311,  483,  621,  193,  272,  48l,  and  373. 

19.  613,  481,  473,  48,  482,  491,  and  453. 

20.  19},  444ft,  737},  and  385}. 

21.  A  farmer  sold  317}}  bushels  wheat,  176}}  bushels  timothy 
seed,  202}}  bushels  buckwheat,  526ff  bushels  corn,  1 75 1|  bushels 
oats,  and  276|f  bushels  clover  seed.     How  many  bushels  did  he 
sell  altogether  ?     (See  Note,  Art.  208.) 

22.  A  jeweler  has  nine  diamonds  whose  respective  weights 
expressed  in  carats  are  }},  |,  ft,  },  }},  },  -},  #,  }}.      Find  their 
total  weight. 

#3.  How  many  inches  of  moulding  would  be  required  for  three 
frames  whose  dimensions  are  as  follows :  first,  11}  inches  wide, 
17}  inches  long  ;  second,  18}  inches  wide,  26}  inches  long  ;  third, 
17}  inches  wide,  24}  inches  long  ? 


64  FRACTIONS.  [Ait.  210 


SUBTRACTION     OF     FRACTIONS. 

210.  Subtraction  of  Fractions  is  the  process  of  finding 
the  difference  between  two  fractions. 

211.  PRINCIPLE. — In  order  that  fractions  may  be  subtracted, 
they  must  have  like  denominators  and  be  parts  of  like  units. 

Ex.    From  f  take  |. 

OPERATION.  ANALYSIS. — As  these  fractions  have  a  common 

-J  —  f  =  |  =  i         denominator,  we  take  the  difference  between  the 
numerators,   and  place  it  over   the   common  de- 
nominator.    |  =  |  is  the  result  required. 

Ex.    What  is  the  difference  between  f  and  f  ? 

OPERATION.  ANALYSIS. — Reduce  the  given  fractions 

9  —  8  to  equivalent   fractions    having    the    least 

4        f  —      -jlj —  **         common  denominator  (1O7).    Then  proceed 
as  in  the  previous  example. 

Ex.    From  176f  subtract  89f . 

OPERATION. 

176$         -f  ANALYSIS. — |  from  f  we  cannot  take  ;  we  therefore 

§93.         &.  take  1  =  f  from  176,  leaving  175.     f  +  f-  =  -y-.     V~  ~  f 

=  f.     175  -  89  =  86.     86  +  f  =  86f . 


5.  RULE. — Reduce  the  given  fractions  to  equivalent 
fractions  having  the  least  common  denominator.  Write 
the  difference  between  the  numerators  over  the  common 
denominator,  and  reduce  the  resulting  fraction  to  its 
simplest  form. 

When  there  are  mixed  numbers,  subtract  the  integers 
and  fractions  separately,  and  add  the  results. 

EXAM  PLES. 

213.  Find  the  difference  between 

1.  |  and  f .  4.     24  and  1^.  7.     1J  and  f. 

2.  |  and  •&.  5.     -fa  and  ^.  8.     f  and  ^-. 

3.  |  and  U.  6.     4  and  4.  9.     1  and  U. 


Art.  213.]      MULTIPLICATION    OF    FRACTIONS.  65 

X 

Find  the  difference  between 
10.  17*  and  9J.  17.  116$  and  48|.  ££.   764J  and  375^. 


11.  175^  and  86J.   7<9.  381f  and  17}.  25.  827-J-  and  737f. 

18.  138|  and  17f   19.  157f  and  19}.  26.  919}  and  447TV 

IS.  149£  and  18f  ^^1183  and  482.  27.  3761  and  2873. 

14.  416|  and  49J.XJW.  387f  and  116}.  28.  4452  and  3183. 

15.  512£  and  53f   22.  248^  and  129£.  #0.  7373  and  4382. 

16.  100  and  13J.   23.  764^  and  375|.  80.  6481  and  5263. 

MULTIPLICATION     OF     FRACTIONS. 
To  multiply  a  fraction  by  an  integer. 


PRINCIPLE. — Multiplying  the  numerator  or  dividing  the 
denominator  by  a  number  multiplies  the  value  of  the  fraction  by 
that  number  (183,  1). 

Ex.    What  will  4  pounds  of  tea  cost  @  $J  a  pound  ? 

OPERATIONS.  ANALYSIS. — If  1  pound  costs  $£, 

4x7  4  pounds  will  cost  4  times  $£,  or 

g '    a    :  ~  *\        $-8/-,  equal  to  $3£.    Hence,  4  pounds 

of  tea  @  $|  will  cost  $3£. 

Or,  To  multiply  £  by  4,  multiply  the 

7  numerator  7  by  4,  or  divide  the  de- 

*  *         g  -JL_  4  ~ "    2      -  «*  y        nominator  8  by  4  ;  either  operation 

will  give  3^,  the  required  product 
Or>  (Prm.). 

-^X^  =  ]£  =  3^  By     cancellation    (166),     the 

2  operation  is  shortened,  and  the  re- 

sult is  obtained  in  its  lowest  terms. 

Multiplying  the  numerator,  as  in  the  first  operation,  increases  the  num- 
ber of  parts,  their  size  remaining  the  same  ;  dividing  the  denominator  multi- 
plies the  fraction  by  increasing  the  size  of  the  parts,  their  number  remaining 
the  same. 

Ex.    Multiply  123f  by  9. 

OPERATION. 

ANALYSIS. — Multiply  the  fraction  f  and  the  integer  123 

separately,  and  add  the  products.     In  practice,  when  possible, 

53         add  the  products  mentally  ;  e.  g.,  9  times  fare  -2^,  equal  to  6|. 

1107  Write  the  f.    9  times  3  are  27,  and  6  are  33.    Write  the  3, 

carry,  and  proceed  as  in  simple  numbers. 


6G 


FRA  CTIONS. 


[Art.  215, 


Ex.    Multiply  227}  by  175. 


ANALYSIS. — As  in  preceding  ex- 
ample. 

Or,  by  aliquot  parts,  when  the 
fractions  are  fourths,  eighths,  etc., 
the  fractions  generally  used  in  com- 
mercial operations. 

*  =  *  +  *(*  of  *). 

£  of  175  =  87^ 
\  of  175,  or  |  of  87|  =  43f . 


216.  RULE. — Multiply  the  numerator  or  divide  the  de- 
nominator of  the  fraction  by  the  integer. 

When  the  multiplicand  is  a  mixed  number,  multiply 
the  fraction  and  integer  separately,  and  add  the  results. 


OPERATIONS. 

227}    Or,    227} 
175          175 

4  )  525 

131J 

1135 

1589 

227 
39856J 

87|- 
44f 

1135 

1589 

227 

39856i- 

EXAMPLES. 

1.  Find  the  cost  of  20  yards  of  silk  at  $J  a  yard. 

2.  How  much  grain  in  12  bins,  each  containing  76-J  bushels  ? 

3.  If  1  man  earns  $J  in  1  day,  how  much  will  16  men  earn  in 
26  days  ? 

4.  If  a  ton  of  hay  cost  $16},  how  much  will  22  tons  cost  ? 

5.  Required  the  cost  of  60  yards  of  muslin  at  35|  cents  a  yard. 

Multiply 

S.  ^  by  7.  17.  412f  by  47. 

7.  -B-byS.  18.  148|by40. 

8.  1%  by  3.  19.  412f  by  89. 

9.  110 J  by  12.  20.  775|  by  65. 

10.  117}  by  16.  21.  119T\  by  20. 

11.  248|  by  3.  22.  772}  by  17. 

12.  146f  by  3.  23.  338|  by  30. 
IS.  1971-  by  7.  24.  550|  by  27. 
j£  420^  by  8.  £5.  643}  by  121. 
15.  384f  by  12.  26.  875 }  by  234. 
Jf0.  375£  by  48.  07.  916£  by  275. 


28.  234|by318. 

29.  678f  by  427. 

50.  625}  by  516. 

51.  7181  by  542. 

32.  275|  by  287. 

33.  813|  by  319. 
84.  444Jby412. 

35.  555f  by  875. 

36.  81 7}  by  416. 

37.  913£  by  375. 

38.  787|  by  525. 


Art.  218.]      MULTIPLICATION    OF   FRACTIONS, 


6? 


218.  To  multiply  an  integer  by  a  fraction,  or  to  frnd 
a  fractional  part  of  an  integer. 

219.  PRINCIPLE.  —  Multiplying  by  a  fraction  is  talcing  such 
part  of  the  multiplicand  as  the  fraction  is  of  a  unit. 

Ex.  If  1  ton  of  hay  cost  $18,  what  will  }  of  a  ton  cost  ? 

Or, 


OPERATIONS. 


4  )18 


Or, 

18 
3 


of  ^  =  V  = 


ANALYSIS.— If  1  ton  cost  $18,  f  of  a  ton  will  cost  f  of  $18.  f  of  $18  is 
3  times  £  of  $18.  £  of  $18  is  $4£  (taking  £  is  the  same  as  dividing  by  4),  and 
3  times  $4|  is  $13£. 

Or,  f  of  $18  is  |  of  3  times  $18.     3  times  $18  is  $54.     £  of  $54  is 

Ex.    Find  the  product  of  175  and  8|. 


OPERATIONS. 

175         Or,         175 


4)  525 


43* 
3 

131} 

1400 


1400 
1531} 


Ex.    Multiply  275  by  47|. 


ANALYSIS. — Multiply  by  the  frac- 
tion f  and  by  the  integer  8  separately, 
and  add  the  products. 

The  first  method  is  preferable, 
when  the  denominator  of  the  fraction 
is  not  an  exact  divisor  of  the  multipli- 
cand. 


FIRST 
OPERATION. 


SECOND  THIRD 

OPERATION.          OPERATION. 


275 
_47f 
8)  825 

103J 

1925 
1100 
13028J 

275 
47| 

275 
_47| 
68J 
84| 
1925 
1100 
13028| 

34| 
3 

103J 
1925 
1100 

ANALYSIS.—  For  the  first  and 
second  operations,  as  in  the  pre- 
ceding examples. 

When  the  fractions  are 
fourths,  eighths,  etc.,  multiply 
by  means  of  aliquot  parts. 


|  of  275  -  68f. 
|  of  275,  or  i  of  68|  =  34|. 


68  FRACTIONS.  [Art.  220. 

220.  KULE. — Multiply  by  the  numerator  of  the  fraction 
and  divide  the  product  by  the  denominator.     Or, 

Divide  by  the  denominator  of  the  fraction  and  multi- 
ply the  quotient  by  the  numerator. 

When  the  multiplier  is  a  mixed  number,  multiply  by 
the  fraction  and  integer  separately,  and  add  the  results. 

EXAM  PLES. 

221.  1.  Find  the  cost  of  8|  yds.  of  ribbon  at  25  cts.  a  yard. 

2.  What  is  the  cost  of  42-J  pounds  of  butter  at  26  cts.  a  pound  ? 

3.  Required  the  value  of  48^  yards  of  flannel  at  75  cts.  a  yard. 

Multiply 

£     84byf.  10.  216byl4|.  16.    780  by  64f . 

5.  126  by  4- .  11.  375  by  24f .  17.   512  by  37f 

6.  49  by  £.  12.  375  by  22f .  18.   611  by  87f 

7.  128  by  9J.  13.   146  by  28}.  19.    625  by  92|. 

8.  156  by  8J.  ^.   184  by  16f.  20.   937  by  75}. 

9.  187  by  10}.  75.   110  by  41}.  &?.   575  by  81}. 

222.  To  multiply  a  fraction  by  a  fraction. 

Ex.    At  I-J  a  pound,  what  will  f  of  a  pound  of  tea  cost  ? 

OPERATION. 

37  £  i   7  ANALYSIS. — If  1  pound  cost  $f  f  of  a  pound 

will  cost  f  of  |f    |  of  $|  is  3  times  £  of  |f 
Or,      }  X  J  =  A        i  of  $£  is  |^,  and  3  times  $^  is  |ft,  or  $^. 

3 

Ex.    What  is  the  value  of  8  x  8|  x  T"%  X  ^  ? 

2  OPERATION.  ANALYSIS. — Reduce  the  inte^ 

4          5^  ger  8  and  the  mixed  number  8^ 

-  -3    x  ^  x  TS  -  :  V  :  =  3i         to  improper  fractions,  and  mul- 
tiply as  in  the  preceding  example. 

223.  KULE. — Reduce  integers  and  mixed  numbers  to  im- 
proper fractions. 

Cancel  oil  factors  common  to  the  numerators  and  de- 
nominators. 

Multiply  the  remaining  numerators  together  for  the 
numerator,  and  the  remaining  denominators  for  the  de- 
nominator. 


Art.  224.]        MULTIPLICATION    OF   FRACTIONS.  69 

EXAMPLES. 

224:.    Find  the  product  of 

1.  I  and  f .  5.  I  and  Y-  9.  *,  13*,  and  *. 

&  |  and  f  5.   6,  3*,  and  f  JO.  26},  4,  and  f . 

5.  f  and  T5f.  7.  of,  4,  and  f}.  JJ.  },  $,  and  164. 

4-  f  and  ^.  8.  12J,  lOf,  and  A«  ^.   13*,  f,  and  -J. 

Reduce  the  following  compound  fractions  to  simple  ones. 

A  Compound  Fraction  is  a  fraction  of  a  fraction. 
The  word  "  of  "  is  equivalent  to  the  sign  x  . 

18.  i  of  f  of  f .  17.  |  of  |  of  18.  21.  I  of  ff  of  }J. 

14.  |  of  34  of  f  18.  |  of  11|  of  4.  22.  f  of  }  of  |  of  -f. 

15.  |  of  |  of  7*.  19.  *-}•  of  ff.  #5.  }  of  12J  Of  6|. 

16.  }  of  }  of  5J.  ^.  T7¥  of  |  of  T5¥.  ^.  |  ofjfaei:  4}. 

Find  the  value  of  the  following  expressions  : 

25.  |  of  1728.  50.  ($  -f  ^)  x  (f  +  A)- 

^.  |  x  375.  81.  (}  -  f )  X  (1  +  }). 

27.  I  times  864.  82.  (A  +  I)  x  (y5¥  —  |). 

^<9.  f  of  75  x  f  of  16|.  88.  37-J  times  |  of  TV 

29.  I  x  }  of  A  x  |.  54.  |  of  |  x  |  of  f. 


To  multiply  mixed  numbers  together. 

Ex.    Multiply  147}  by  41f 

OPERATION.  ANALYSIS. — Commencing  at  the 

147f  right  as  in  multiplication  of  integers, 

41 5  first  multiply  the  fraction  and  inte- 

ger of  the  multiplicand  by  the  frac- 

tf    :         "s   x  T  tion  of   the    multiplier  ;    and   then 

8  )  735  91J      =      |  X  147         multiply  the  fraction  and  integer  of 

4  )  123  30f      =    41   X  f  the  multiplicand  by  the  integer  of 

•j^    -v  the  multiplier.      The  separate  steps 

j-  =    41   X  147         are   indicated   at  the   right   of  the 

operation.     The  sum  of  the  several 

6150^-  partial  products  will  be  the  required 

product. 


70 


FRA  CTIONS. 


[Art.  221 


226.  BULK — Multiply  the  fractions  tog^her,  each  integer 
by  the  fraction  of  the  other  number,  and  the  integers  to- 
gether. The  sum  of  the  partial  products  will  be  the  product 
required. 

NOTE. — When  both  mixed  numbers  are  small,  reducfe  them  to  improper 
fractions  and  multiply  as  in  multiplication  of  fractions  (2 


EXAM  PLES. 

221.  Multiply                             /Multiply 

1. 

875£  by    8J  ;  by  37  *. 

/    6.     H6f  by  27J-  ; 

by  581. 

2. 

737i  by  10£  ;  bv  12J.           i 

f     7.     447Jby45i; 

by  641. 

S. 

512}  by    7J  ;  by  27f 

8.     459}  by  37}  ; 

by  39} 

k 

449f  by  16i  ;  by  36f 

1      9.     378jby43i; 

by4aj( 

5. 

612J  by  13J  ;  by  42|. 

\  10.     479£  by  56J  ; 

by  2/f  . 

DIVISION     OF 

FRACTIONS. 

22S.   To  divide  a  fraction 

by  an  integer. 

PRINCIPLE. — Dividing  the  numerator  or  multiplying 
the  denominator  by  a  number,  divides  the  value  of  the  fraction  by 
that  number  (183,  2). 


Ex.    "What  cost  1  pound  of  tea,  if  5  pounds  cost 

OPERATIONS.  ANALYSIS. — If    5    pounds    cost 

$3£,    1   pound  will  cost  £  of  $&*, 
or$|. 

To  divide  ^  (3$)  by  5,  divide 
the  numerator  10  by  5,  or  multiply 
the  denominator  3  by  5  ;  either 
operation  will  give  f,  the  required 
quotient  (Prin.). 

Dividing  the  numerator,  as  in 

the  first  operation,  decreases  the  number  of  parts,  their  size  remaining  the 
same  ;  multiplying  the  denominator  divides  the  fraction  by  decreasing  the 
size  of  the  parts,  their  number  remaining  the  same. 


Or, 


Ex.    Divide  867f  by  4. 


OPERATION. 


4  )  867|         3f  =  V- 
216||       if.  .=.  4  = 


ANALYSIS. — Dividing  as  in  simple 
numbers,  4  is  contained  in  867f,  2  HI 
times  and  a  remainder  of  3f .  3f  equals 
-\5-,  which  divided  by  4  is  ||. 


Art.  330.] 


DIVISION     OF    FRACTIONS. 


71 


23O.  RULE. — Divide  the  numerator  or  multiply  the 
denominator  of  the  fraction  by  the  integer. 

When  the  dividend  is  a  mixed  number,  divide  the 
integer  and  the  fraction  separately,  and  add  the  results. 


231. 

x~*^S 

Divide 

/" 

1. 

fby 

6. 

fn. 

2. 

fby 

3. 

/      12. 

3. 

fby 

6. 

1      18. 

4. 

A  by  4.          /       14. 

5. 

A  by  4. 

15. 

6. 

16J-  by  5. 

16. 

7. 

172J 

by  3. 

17. 

8. 

875f 

by  6. 

18. 

9. 

0. 

935f 
729J 

by  8. 
by  9. 

\       19. 
\    20. 

21. 

22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 

5316J 
7144J 
1729J 
1749J 
8763J 
7385J 
4255| 

7134 

9727| 
6345| 

by 
by 
by 
by 
by 
by 
by 
by 
by 
by 

4, 
5. 
3. 

9. 
6. 
8. 
9. 
7. 
12. 
16. 

EXAMPLES 

637£  by  9. 
875TV  by  12. 
1716f  by  8. 
1729£  by  3. 
241 8  J  by  5. 
351 6f  by  5. 
2428J  by  3. 
6375|  by  4. 
4287|  by  2. 
3281^  by  8. 


232.  To  divid£\by  a  fraction. 

233.  The  Reciprocal  of  a  number  is  1  divided   by  that 
number.     Thus,  the  reciprocal  of  4  is  1  divided  by  4,  or  J. 

The  Reciprocal  of  a  Fraction  is  1  divided  by  that  fraction, 

234.  PRINCIPLE.  —  1  divided  by  a  fraction  is  the  fraction  in- 
verted. 

Thus,  1  divided  by  f  is  |.  This  principle  may  be  demonstrated  as  fol- 
lows :  In  1  there  are  4  fourths.  1  fourth  is  contained  in  4  fourths  4  times. 
Since  f  is  3  times  \,  f  is  contained  in  1  ^  as  many  times  as  ^.  Hence,  £  is 
contained  in  1  ^  of  4  times,  or  f  times. 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 

Ex.  At  $f  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $5  ? 

OPERATIONS.  ANALYSIS.  —  Since    1    yard    cost 

5  _._  3.  —  2_o  _._  i  _  6  2  $f,  as  many  yards  can  be  bought  for 

$5  as  $f  is  contained  times  in  $5. 
Or,   5  -f-  f  =  =  -f-  X  f  =  :  -V-  =  =  6|       5  ig  equal  to  ^  and  3  fourths  is  con- 


Or, $f  is  contained  in  $1 
equal  to  6f  times. 


tained  in  20  fourths  6|  times. 
times  (Prin.\  and  in  $5,  5  times  |  or 


72  FRACTIONS.  [Art.  234. 

Ex.    At  If  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  If-  ? 

OPERATIONS.  ANALYSIS.  —  Since  1  yard 

I  _:-  J  =  i-|  -:-  T9^  —  1£  cost  $|,  as  many  yards  can  be 


Or      JL       ^  -  5   x  4  _  A  o  =  ii  or        as         s  con- 

tained  times  in  $f  .    f  is  equal 

Or        5    •    3   -   8    v  4-  -_   10   -  11          to  A'  and   *  is  e(lual  to  "• 
^      IT  —  4  -  •  -f  '  ^  is  contained  in  }$  1£  times. 

Or,  $|  is  contained  in  $1 
|  times  (Prin.\  and  in  $f,  f  times  £  or  f£,  equal  to  1£  times. 

Ex.    If  6f  yards  of  cloth  cost  $5,  what  will  1  yard  cost  ? 

OPERATIONS.  ANALYSIS.  —  6f  yards  are 

5  ^_  zg.  —  (5  H-  20)  X  3  =  I        equal  to  *£  yards.     Since  -3^ 

Or,     5  —  AA  =  4-xA  =  U=:'|-       yards  cost  $5'  *  of  a  yard 

will  cost  ^  of  $5  or  $i,  and 
Or,     5  -T-  Y  =  f  X  tfc  ==  f  3  or  j  yard  will  cost  3  times 

4  $J  or  $f. 

Or,  the  price  per  yard  equals  the  cost,  divided  by  the  quantity  as  an  ab- 
stract number.  5  divided  by  %°-  equals  5  times  1  divided  by  -85°-,  or  5  times  ¥% 
(Prin.),  equal  to  f. 

Ex.    Divide  7552  by  78f. 

7-K9  ANALYSIS.  —  Reduce  both  divisor  and  dividend 

to    thirds    as    in    the    operation,    omitting    the 

3  ___  3  common  denominators.     *i«fi«.  -s-  £|«.  js  ^e  same 


236)  22656  (96         as  22656  ^236. 

21  y.  Or,  multiplying  both  divisor  and  dividend  by 

the  same  number  does  not   affect  the  quotient. 
1416  Multiply  both   divisor  and  dividend  by  3,  and 

then  divide  as  in  simple  numbers. 


Ex.    Divide  2195$  by  175|. 

OPERATION. 

1751  )  2195| 


f.  ANALYSIS.  —  Reduce  both  divisor  and  divi- 

dend to  sixths,  their  least  common  denomina- 


1054  )  13175  (  12£  tor,    reject    the    common    denominator,    and 

1054.  divide  the  numerators  as  in  simple  numbers. 

Or,  multiply  both  divisor  and  dividend  by 
6,  the  least  common  denominator,  and  divide 
as  in  simple  numbers  (see  preceding  analysis). 
j>21  175|  =  175f. 

1054  ~~  * 


Art.  235.] 


DIVISION     OF    FRACTIONS. 


235.  RULE. — Reduce  the  divisor  and  dividend  to  equiv- 
alent fractions  having  a  common  denominator,  and  divide 
the  numerator  of  the  dividend  by  the  numerator  of  the 
divisor.  Or, 

Invert  the  terms  of  the  divisor  and  proceed  as  in  multi- 
plication. 

In  dividing  mixed  numbers,  multiply  both  divisor  and 
dividend  by  the  least  common  denominator,  and  divide  as 
in  simple  numbers. 

NOTE. — If  both  mixed  numbers  are  small,  reduce  them  to  improper  frac- 
tions, and  apply  the  rule  for  division  of  fractions. 


EXAMPLES. 


236.    Divide 
1.     Ibyf 
16  by  f . 

28  by  f . 
49  by  J. 
88  by  f . 
fbyf 


2. 
3. 

4- 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 


A  by  i. 

i  a,      Jo 

fbyf. 
28  by  4£. 
33  by  3f . 
64  by  5f . 


27. 


875 
625  by  83  \. 
516  by  34f . 
917  by  43f . 

864  by  86f . 
702  by 


920  by  73f. 
720  by  43£. 
700  by  37£. 
560  by  26J. 
682J  by  45J. 
847!  by  89£. 
9843  by  75^. 
86&?  by  183. 
7311  by  56i. 
431J  by  18}. 
983i  by  29f 
504|  by  36f . 
5831  by  43f . 


Find  the  value  of  the  following  complex  fractions  (182)  and 
expressions  of  division  : 


40. 


|A  .  4| .  24f 
9    '  35 '    36 


/./      _1  •   _1  •  _  1, 
40  '  13  '    20 

H  .   f  .  A 


74 


FRA  CTIONS. 


[Art.  237. 


11}  cts.  per 


REVIEW     EXAMPLES. 

237.     1-  Reduce  f  f  f  to  its  lowest  terms. 

Reduce  -J  to  forty-eighths. 
j       3.  Reduce  727|  to  an  improper  fraction. 
\      4»  Reduce  ^jp-1  to  a  mixed  number. 
\J^&dd  17i,  37},  18f ,  49*,  13|,  and  56^. 
1 6.  From  1728|  take  865}. 

7.  Multiply  ixSJx^x^x  16|. 

8.  Multiply  1727}  by  175. 
f      9.  Multiply  1727  by  175}. 

10.  Divide  1J  by  -£%. 

11.  Divide  1736  by  144f. 

12.  Divide  5779|  by  275f. 

,     IS.  Divide  12346J  by  7  ;  by  35. 

\    14.  -What  is  the  cost  of  1583  pounds  sugar 

p<W<l? 

15.  Add  |  of  I  of  4f,  |,  136f,  and  5*. 

A  merchant  sold  a  quantity  of  goods  for  $144,  which  was 
of  the  cost.     What  was  the  cost  ? 

ANALYSIS. — If  $144  is  f  of  the  cost,  |-  of  the  cost  is  £  (\  is  ^  of  f)  of  $144, 
or  $48.    £,  or  the  total  cost,  is  4  times  (f  is  4  times  {}  $48,  or  $192. 

.77.  Required  the  value  of  2993  pounds  of  sugar  @  9|  cts.  per 
pound  ? 

18.  If  I  of  a  ship  is  worth  $42430J,  what  is  the  value  of  the 
whole  ? 

19.  Bought  47}  yards  of  cloth  at  $4£  per  yard,  and  paid  for  it 
in  wheat  at  $2|  per  bushel ;  how  many  bushels  were  required  ? 

20.  Find  the  value  of  3 Iff  pounds  snuff  @  72  cts.  per  pound. 

21.  The  less  of  two  numbers  is  777}  and  their  difference  117|; 
what  is  the  greater  number  ? 

22.  A  and  B  together  have  $1728  ;  if  A's  money  is  equal  to  -J 
of  B's,  how  much  has  each  ? 

23.  A  having  2146}  yards  of  cloth,  sold  |  of  it  at  $1}  a  yard, 
and  the  remainder  at  $2£  a  yard  ;  how  much  did  he  receive  ? 

24.  A  number  being  increased  by  f  of  itself,  the  sum  is  546  ; 
kwhat  is  the  number  ?     (The  number  is  f  of  itself. ) 


Art.  237.]  REVIEW     EXAMPLES.  75 

25.  A  man  had  $5280  ;  he  bought  goods  with  f  of  it,  and  then 
lent  £  of  the  balance  to  a  friend  ;  how  much  had  he  left  ? 

26.  Find  the  selling  price  of  goods  sold  at  a  profit  of  $75, 
being  ^  of  the  cost. 

27.  Mr.  A  bought  117 J  acres  of  land  at  one  time,  and  87|  at 
another  ;  after  selling  110J  acres,  how  much  remained  ? 

28.  If  8f  tons  of  coal  cost  $30£,  what  will  27-J-  tons  cost  ? 
How  many  tons  can  be  bought  for  $127J  ? 

29.  A  man  paid  $1145f  for  a  horse  and  carriage.     What  was 
the  value  of  each,  the  carriage  being  valued  at  -|  as  much  as  the 
horse  ? 

30.  If  J  of  a  farm  is  valued  at  $2253 1,  what  is  the  value  of  f 
of  it? 

31.  What  is  the  value  of  21021  yards  prints  at  72  cents  per 
yard? 

82.  What  number  must  be  taken  from  96J,  and  the  remainder 
multiplied  by  16|,  that  the  product  shall  be  770f  ? 

33.  What  is  the  value  of  1642  yards  muslin  at  5J  cents  per 
yard? 

34.  If  7  barrels  of  oil  contain  313^  gallons,  how  many  gallons 
will  2J  barrels  contain  ? 

85.  An  executor  collects  $12724.84.     He  pays  out  $4096.48, 
and  the  residue  he  pays  to  the  widow  and  her  four  children  as 
follows  :   The  widow  receives  a  third  part,  and  the  remainder  is 
divided  equally  among  the  children.     Find  the  share  of  each. 

86.  What  number  increased  by  f  of  itself  will  produce  2456£  ? 

37.  Find  the  selling-price  of  goods,  bought  at  $144,  and  sold 
at  ^  above  cost.   , 

38.  A  invests  f  of  his  capital  in  real  estate,  and  has  $1725 
remaining  ;  what  is  his  capital  ? 

89.  Bought  a  barrel  of  sugar  containing  220  Ibs.,  at  8£  cents 
per  pound.  During  the  sale,  it  dried  away  -fa.  Did  I  gain  or 
lose,  and  how  much,  by  selling  it  at  9J  cents  per  pound  ? 

40.  Multiply  2375J  by  8J  ;  by  10J. 

41.  Multiply  1727J  by  18J-  ;  by  107|. 
4$.  Multiply  377J  by  16J- ;  by  37f. 

43.  Multiply  875J  by  22£  ;  by  9|. 

44.  A  merchant  sold  12|  yards  of  silk  to  one  customer,  21f  to 
another,  20|  to  another,  and  28£  to  another ;  at  $2f  per  yard, 
how  many  dollars  did  he  receive  ? 


FRACTIONS.  [Art.  237. 

45.  An  army  loses  -£$  of  its  number  in  battle  and  has  16043 
laining  ;  how  many  did  it  originally  contain  ? 

46.  What  is  the  cost  of  34  pieces  prints,   containing  16042 
,  at  51  cents  per  yard  ? 

47.  What  is  the  value  of  12  pieces  prints  containing  48,  481, 
482,  48,  492,  483,  48,  493,  492,  483,  492,   48s  yards  respectively   at 
43  cents  per  yard  ? 

48.  A  merchant  purchased  29  pieces  prints   containing   483, 
482,  412,  482,  483,  47,  49,   492,  521,   573,  483,  482,    38,    482,    482, 
482,  473,  482,  48,  51,    48,    441,    512,    48,    423,    462,    48,    482,    48s 
yards  respectively  ;  what  was  the  cost  at  52  cents  per  yard  ? 

49.  There  are  5280  feet  in  one  mile,  and  16£  feet  in  one  rod  ; 
how  many  rods  in  one  mile  ? 

50.  A  can  do  a  certain  piece  of  work  in  10  days,  and  B  can  do 
in  15  days ;  how  long  will  it  take  them  both  to  do  it  ? 

A  market-woman  bought  120  oranges  at  the  rate  of  5  for 
Scents,  and  sold  £  of  them  at  the  rate  of  3  for  1  cent,  and  the 
remainder  at  the  rate  of  2  for  1  cent.  Did  she  gain  or  lose,  and 
how  much  ? 

52.  What  is  the  duty  on  22375  pounds  sugar  at  2^-J  cts.  per 
pound  ? 

58.  A  farmer  sold  1276^£  bushels  oats  at  44  cts.  per  bushel, 
876ff  bushels  corn  at  52f  cts.,  and  3381£f  bushels  wheat  at  $1.32; 
how  much  did  he  receive  ? 

54..  How  many  bushels  of  corn  at  54J  cts.  per  bushel  must  a 
farmer  exchange  for  62  yards  of  sheeting  at  8|  cts.  per  yard,  and 
>  3Lyards  broadcloth  at  $1.75  per  yard  ? 

V.       5$\  What   is   the  value  of  453  yards  damask  at    772  cts.  per 
y£r44^ 

56.  The  salary  of  the  President  of  the  United  States  is  $50000 
per  year ;  how  much  is  that  per  day  ? 

57.  l-/g-  pounds  of  beef  and  1T6^  pounds  of  flour  are  allowed  to 
a  ration ;  how  much  will  617  rations  cost,  if  the  price  of  beef  is 
llf  cts.  per  pound,  and  of  flour  3J  cts.  per  pound  ? 

58.  What  is  the  value  of  36385  pounds  of  corn  at  48f  cents 
per  bushel,  each  bushel  containing  56  pounds  ? 

59.  What  is  the  least  common  multiple  of  the  nine  digits  ? 

60.  The  'total   production  of   gold  and  silver  in  the  United 
States  from   1792  to  1886  was   $2,403,986,769.     What  was  the 
average  production  per  year  ? 


DECIMALS. 


238.  A  Decimal  (from  the  Latin  decem,  ten)  Fraction  is 
a  fraction  whose  denominator  is  1  followed  by  one  or  more  ciphers; 
as    ilFJ  TolT?  Tooo?   10060' 

239.  Decimal  fractions  arise  from  dividing  a  unit  into  10 
equal  parts,   and  then  dividing  these  parts  into  10  other  equal 
parts,  and  so  on. 

Thus,  if  a  unit  be  divided  into  10  equal  parts,  each  part  is  called  a  tenth. 
If  a  unit  be  divided  into  100  equal  parts,  or  1  tenth  into  10  equal  parts,  the 
parts  are  called  hundredths.  If  a  unit  be  divided  into  1000  equal  parts,  or 
1  hundredth  into  10  equal  parts,  the  parts  are  called  thousandths. 

240.  All  the  rules,  principles,  operations,  etc.,  of  common 
fractions  may  be   applied   to  decimal  fractions.     Since  decimal 
fractions  increase  and  decrease  uniformly  according  to  the  scale 
of  ten,  a  more  simple  notation,  similar  to  that  of  integers,  has 
been  devised  for  them. 

A  hundred  is  written  100  ;  a  tenth  part  of  a  hundred  (ten)  is  written  10, 
the  1  being  written  one  place  to  the  right  ;  a  tenth  part  of  one  ten  (one  unit) 
is  written  1,  the  1  being  written  one  place  to  the  right  ;  in  like  manner,  a 
tenth  part  of  one  unit  (one-tenth)  is  written  .1,  the  1  being  written  one  place 
to  the  right  ;  the  tenth  part  of  one-tenth  (one  hundredth)  is  writen  .01,  the 
1  being  written  one  place  to  the  right,  etc. ,  etc. 

Decimal  fractions,  like  integers,  decrease  from  left  to  right  in  a  tenfold 
ratio,  and  increase  from  right  to  left  in  the  same  ratio. 

241.  In  the  decimal  notation,  the  numerator  only  is  written, 
the  denominator  being  indicated  by  the  position  of  a  point  (  .  ) 
called  the  decimal  point.     The  decimal  point  separates  the  inte- 
gral from  the  fractional  part. 


78  D  E  CIMA  LS.  [Art.  242. 

24:2.  The  denominator  of  a  decimal  fraction  is  understood, 
and  is  1  with  as  many  ciphers  annexed  as  there  are  figures  in  the 
decimal ;  thus, 

Form  of  Form  of 

common  fraction.  decimal  fraction. 

TV     is  written     .7     and  is  read  seven  tenths. 
Tg-o    "        "          .08       "         "      eight  hundredths. 
yUcr  "        "         -016     "         "      sixteen  thousandths. 

Hereafter,  the  first  form,  that  of  the  common  fraction,  will  be  called  a 
fraction,  and  the  second,  that  of  the  decimal  notation,  a  decimal. 

24:3.  The  first  place  to  the  right  of  the  point  is  called  tenths, 
the  second  place  hundredths,  the  third  place  thousandths,  and 
BO  on. 

244.  The  relation  between  integers  and  decimals  is  shown  in 
the  following 

NUMERATION  TABLE. 


2436807593.  689460582 

S  I  3o  §  I  I  I!  s  s  3      I  s  3  5  I  I  £  i  § 

Orders  of  Integers.  Orders  of  Decimals. 

245.  In  the  above  table,  observe  that  the  first  place  to  the 
left  of  units  is  called  tens,  and  the  first  place  to  the  right,  tenths  ; 
the  second  place  to  the  left  of  units  is  called  hundreds,  and  the 
second  place  to  the  right,  hundredths,  etc.     Hence  the  number 
of  any  order  or  place  of  the  decimal,  counting  from  the  point,  or 
from  units'  place,  is  the  same  as  the  number  of  ciphers  in  the 
denominator  of  the  decimal. 

246.  A  Complex  Decimal  has  a  fraction  in  its  right-hand 
place. 


Thus,  .16f  r-^j  is  a  complex  decimal,  and  is  read  16f  hundredths,  the 
fraction  not  being  counted  as  a  decimal  place. 


Art.  247.]  NUMERATION    OF     DECIMALS.  79 

247.  PRINCIPLES.  —  1.  Annexing  ciphers  to  a  decimal  does  not 
alter  its  value. 

Annexing  a  cipher  multiplies  both  the  numerator  and  the  denominator  by 
10,  and  hence  does  not  alter  the  value  of  the  decimal  (183,  3).     Thus,  .7  (TV) 


2.  Each  removal  of  the  decimal  point  one  place  to  the  right 
multiplies  the  value  of  the  decimal  by  10. 

Removing  the  point  one  place  to  the  right  does  not  change  the  numerator, 
but  divides  the  denominator  by  10,  and  hence  multiplies  the  value  of  the 
decimal  (183,  1).  Thus,  .072  (rffa)  becomes  .72  (flfc)  ;  ^  =  T^  x  10. 

3.  Each  removal  of  the  decimal  point  one  place  to  the  left  divides 
the  value  of  the  decimal  by  10. 

Removing  the  point  one  place  to  the  left  does  not  change  the  numerator, 
but  multiplies  the  denominator  by  10,  and  hence  divides  the  value  of  the  frac- 
tion by  10  (183,  2).  Thus,  .72  (^fr)  becomes  .072  (yflfo);  T£f¥  =  ft,  +  10. 

NUMERATION     OF     DECIMALS. 

248.  KULE.  —  Read  the  decimal  as  if  it  mere  an  integer, 
and  give  it  the  name  of  its  right-hand  order. 

EXERCISES. 

249.  Write  in  words,  or  read  orally  the  following  numbers  : 

1.  .6.  8.  17.6.  15.  375.  18|. 

2.  .008.  9.  8.029.  16.  19.0033J. 

3.  .27.  10.  24.000488.  17.  6.148|. 

4.  .0375.  11.  400.00008$.  18.  648.  6|. 

5.  .0108.  12.  76.7071.  19.  347.18005. 

6.  ."75.  13.  3000.0045.  20.  808.008. 

7.  .1007.  14.  .3045.  21.  600.06. 

NOTATION     OF    DECIMALS. 

250.  Write  sixty-four  thousandths  in  the  form  of  a  decimal. 

ANALYSIS.  —  Since  there  are  only  two  figures  in  the  numerator  64,  and  the 
right-hand  figure  of  the  decimal  must  occupy  the  third  decimal  place  to  ex- 
press thousandths,  it  is  necessaiy  to  prefix  a  cipher  to  bring  the  right-hand 
figure  into  its  proper  place.  Therefore  write  point,  naught,  six,  four  (.064)  in 
the  order  named. 


80  DECIMALS.  [Art.  251. 


RULE.  —  Prefix  the  decimal  point,  and  decimal 
ciphers  if  necessary,  to  the  numerator  written  as  an  integer, 
so  that  the  right-hand  figure  will  occupy  the  order  named. 

NOTE.  —  Before  writing,  determine  mentally  the  place  of  the  right-hand 
figure  and  the  number  of  ciphers  required.  Write  in  all  cases  from  left  to 
right. 

EXERCISES. 

252.  1.  What  is  the  name  of  the  third  decimal  order  ?  The 
.sixth  ?  The  first  ?  The  fourth  ?  The  second  ?  The  seventh  ? 

2.  How  many  decimal  places  are  required  to  express  hun- 
dredths?  Millionths  ?  Ten-thousandths?  Tenths?  Hundred- 
millionths  ?  Hundred-thousandths  ? 

8.  How  many  ciphers  must  be  written  after  the  decimal  point 
in  writing  375  millionths  ?  27  hundredths  ?  875  thousandths  ? 
446  ten-millionths  ?  37  ten-thousandths  ? 

4.  Write  the  following  as  decimals,  so  that  the  decimal-points 
stand  in  the  same  vertical  line  :   8  tenths  ;  16  hundredths  ;  175 
thousandths  ;  1804    millionths  ;   56   ten-thousandths  ;  3004  ten- 
millionths  ;  1728  ten-thousandths. 

5.  Seventeen,  and  seventy-five  hundredths. 

6.  Twenty-six,  and  twenty-six  thousandths. 

7.  Two  hundred  forty-six  ten-millionths. 

8.  Two  hundred,  and  forty-six  ten-millionths. 

9.  Three  hundred  seventy-five,  and  eighteen  hundred-thou- 
sandths. 

10.  Eight  thousand,  and  sixty-five  ten-thousandths. 

11.  Eight  thousand  sixty-five  ten-thousandths. 


Art.253.]  REDUCTION     OF    DECIMALS.  81 

REDUCTION     OF    DECIMALS. 

253.  To  reduce  a  fraction  to  a  decimal. 

Ex.    Reduce  f-  to  a  decimal. 

OPERATION. 

4.  \   3  00  ANALYSIS. — f  equals  £  of  3  units.     3  units  equal  300 

hundredths.     £  of  300  hundredths  equal  75  hundredths. 

.75 

254.  RULE. — Annex  decimal  ciphers  to  the  numerator, 
and  divide  by  the  denominator,  pointing  off  as  many  deci- 
mal places  in  the  quotient  as  there  are  ciphers  annexed. 

255.  A  fraction  in  its  lowest  terms  can  be  reduced  to  a  pure 
decimal  only  when  its  denominator  contains  no  prime  factors  but 
2  and  5.     If  the  denominator  or  divisor  contain  any  prime  factor 
other  than  2  and  5,  the  divisor  will  not  end.     The  decimals  thus 
produced  are  called   Interminate  or   Repeating   Decimals, 
and  the  figures  repeated,  Repetends. 

When  a  fraction  is  in  its  lowest  terms,  its  numerator  and  denominator 
have  no  common  factors  (187).  Annexing  ciphers  to  the  numerator 
introduces  the  factors  2  and  5  only  ;  hence,  if  the  denominator  is  an  exact 
divisor  of  the  numerator  with  the  ciphers  annexed,  it  must  contain  these 
prime  factors  and  no  others. 

EXAMPLES. 

256.  Reduce  to  equivalent  decimals  : 

1.  i-  4.  f.  7.  H.  10.  TV         .    13.  16f.    - 

5.  TV  8.  f.  11.  f  14.  27|f. 

6.  ff.  9.  f  -  12.  f .  15.  36|f. 

257.  To  reduce  a  decimal  to  a  fraction. 

Ex.    Reduce  .075  to  an  equivalent  fraction. 

ANALYSIS. — A   decimal  is   changed  to  a 

OPERATION.  fraction  by  writing  its  denominator,  and  omit- 

.075  =  yj^  -f-  -fe          ting  the  decimal  point  and  prefixed  ciphers. 

=  A 


82                                                        DECIMALS.  [Art.  257. 

Ex.    Change  .83J  to  a  simple  fraction. 

OPERATION.  ANALYSIS.— Reduce  the 

•83*  =      =          =  «*  =  *  complex  fraction     to ' 


simple   fraction  by  multi- 
plying both  terms   by  the   denominator  3.      (183,  3.) 

258.  RULE. — Qmitthe  decimal  point,  supply  the  proper 
denominator,  and  reduce  the  fraction  to  its  lowest  terms. 

EXAM  PLES. 

259.  Reduce  to  equivalent  fractions : 


1. 

.25. 

8. 

.128. 

15. 

.33^. 

ff)0) 

%rC. 

.44*. 

2. 

.75. 

9. 

.00144. 

16. 

.41|. 

23. 

.142857f 

3. 

.375. 

10. 

.512. 

17. 

.066f. 

24. 

.0833^. 

4- 

.625. 

11. 

.5625. 

18. 

.37f 

25. 

28.0375. 

6. 

.875. 

12. 

.1875. 

19. 

.104£. 

26. 

107.166|. 

6. 

.125. 

13. 

.12|, 

20. 

.097f. 

27. 

175.096. 

7. 

.016. 

14. 

.16|, 

21. 

.0053f 

28. 

.6.0175. 

ADDITION     OF    DECIMALS. 

26O.  Since  decimals,  like  integers,  increase  and  decrease  uni- 
formly according  to  a  scale  of  ten,  with  the  exception  of  placing 
the  decimal  point  in  the  result  (usually  called  pointing  off),  they 
may  be  added,  subtracted,  multiplied,  and  divided  in  the  same 
manner  as  integers. 

Ex.    What  is  the  sum  of  28.7,  175.28,  .037,  25.0045,  and  4.08  ? 

OPEBATION. 

28.7  ANALYSIS. — Write  the  numbers  so  that  units  of  the 

175.28  same  order  stand  in  the  same  column. 

^037  If  the  decimal  points  are  in  the  same  vertical  l\ne. 

25  0045  tenths  will  necessarily  be  under  tenths,  hundredths  under 
hundredths,  etc.    Add  as  in  integers,  and  place  the  point 

^•0°  in  the  result  directly  under  the  points  of  the  numbers. 
233.1015 


Art.  260.]  ADDITION     OF    DECIMALS.  83 

Ex.    Add  .6,  .37|,  16.0484,  8.12344,  and  24.125. 

OPERATION.  ANALYSIS. — Reduce  the  complex  deci- 

.6  =        .6  mals  as  far  as  the  decimal  places  extend 

.37}        — •        .3775  in  the  other  numbers.    Since  the  fractio»s__ 

16.0481      =    16.04831  now  express  parts  of  the  same  fractional 

ft  1  9*A2  ft  1  93A2  unit'  the^  ma^  be  adcle(i- 

O.  1/&O4:*-      :=          O.  JL/wOrHf-  T  .  ,          .  .  . 

7  *  In  practice,  the  fractions  may  be  re- 

:    ^4:- 1%5 jected  if  the   decimals  are   carried  one 

49.2742}f         place,  at  least,  farther  than  accuracy  is 
required. 


RULE. —  Write  the  numbers  so  that  their  decimal 
points  are  in  the  same  vertical  line.  .Add  as  in  integers, 
and  place  the  decimal  point  in  the  result  directly  under 
the  points  in  the  numbers  added. 

EXAMPLES. 


262.  1.  Add  ninety-seven  hundredths ;  three  hundred 
forty-seven  thousandths  ;  sixteen,  and  seventy-five  hundred-thou- 
sandths ;  four  hundred  seventy-five,  and  two  thousand  thirty- 
seven  millionths. 

2.  Add  four,  and  eighty-one  thousandths ;  thirty-seven,  and 
two  hundred  one  ten-thousandths  ;  seven  thousand  eight  hundred- 
thousandths  ;  seven  thousand,   and  eight   hundred-thousandths ; 
nineteen  hundredths  ;  three  hundred  sixty-four,  and  nine  tenths ; 
and  fifty-six,  and  fifty-four  thousandths. 

3.  Add  three   hundred   seventy-five,  and  eight  hundredths ; 
eighteen  thousandths  ;  ninety-six,   and  eighty-four  hundredths ; 
four,  and  four  tenths  ;  and  eight  hundred  seven  ten-millionths. 

4.  What  is  the  sum  of  18  hundredths  ;   716  hundred-thou- 
sandths ;  6342  millionths  ;  11567  ten-millionths ;    625  ten-thou- 
sandths ;  9  tenths  ;  99  hundredths  ;  and  512  thousandths  ? 

^_^Add   81.86;     12.593;    4.004;    18.00129;    .443;    400.043; 
.12875;  175.00175;  17.3008;  9000.0016;  and  .9016. 

6.  Find  the  sum  of  99  ten-thousandths  ;  157}  thousandths ; 
789J  millionths  ;  6  tenths ;  18J  hundredths  ;  1728  ten-millionths ; 
and  88  hundredths. 

7.  Add   $1728.64;    $0.37};    $18.44};    $10.18};    $6.25;   and 


84  DECIMALS.  [\rt.262. 

8.  What  is  the  sum  of  $12.37 J ;  $144.18};  $6.62£;  $175.06^; 
$40.17|;  and  $398? 

P.  Add  .1263$;  12.875;  187.25;  9.1414J;  .12;  5.7604TV; 
and  .0008|. 

10.  Add  .264;  4.18};  .0017f;  .00864$;  .04|;  17.387$;  and 
.0102075. 

SUBTRACTION     OF    DECIMALS. 

263.  Ex.    From  12.75  subtract  8.125. 

OPEBATION.  ANALYSIS. — Write  the  subtrahend  under  the  minuend  so 

12. 75  that  units  of  the  same  order  stand  in  the  same  column.     Sub- 

8.125         tract  as  in  integers,  and  place  the  point  in  the  result  directly 
~~"~         under  the  points  of  the  numbers. 

If,  as  in  this  example,  the  minuend  has  not  as  many  deci- 
mal places  as  the  subtrahend,  suppose  decimal  ciphers  to  be  annexed  until 
the  right-hand  figures  are  of  the  same  order.     (247,  1.) 
Reduce  complex  decimals  as  in  addition  (26O). 

264.  RVLE. —  Write  the  numbers  so  that  their  decimal 
points  are  in  the  same  vertical  line.     Subtract  as  in  inte- 
gers, and  place  the  point  in  the  remainder  directly  under 
the  points  in  the  minuend  and  subtrahend. 

EXAMPLES. 

265.  1.    From   four,   and  sixty-five   thousandths,    subtract 
eight  hundred  forty-seven  ten-thousandths. 

2.  From  twenty-seven  hundredths  take  twenty-nine  hundred- 
thousandths. 

3.  From  nine  thousand,  and  thirty-four  ten-thousandths,  sub- 
tract nine  thousand  thirty-four  ten-thousandths. 

Find  the  difference  between 

4.  8.3644  and  7.8996.  12.  17.864|  and  16.94. 

5.  17.4586  and  .785.  13.  144.43$  and  113.3875. 

6.  1.010101  and  .999999.  14.  54.37|  and  .98f. 

7.  $173.46  and  $87.29.  15.  117.48}  and  49.434. 

8.  3  and  .873845.  16.  448.987$  and  38^28$. 

9.  17.24$  and  18.973}.  17.  5556.8}  and  44.48. 

10.  $510. 60  and  $389. 45$.  18.     968.44$  and  37.386f. 

11.  $1728  and  $.06}.  19.     49.45$  and  48.9876$. 


Art,  266.]         MULTIPLICATION     OF    DECIMALS.  85 


MULTIPLICATION     OF    DECIMALS. 

266.     Ex.    Multiply  '.144  by  .12. 

OPERATION.  ANALYSIS.—  .144  x  .12  =  ^%  x  jfo  =  tfffo.     Multiply 

.144  the  numerators  of  the  two  factors  for  the  numerator  of  the 

.12  product,  as   in   multiplication  of  fractions.     In  the  above 

multiplication   of  fractions,    it   will   be   observed  that  the 
m  *79ft 

number  of  ciphers  in  the  denominator  of  the  product  equals 

the  sum  of  the  ciphers  in  the  denominators  of  the  two 
factors.  Since,  each  cipher  represents  a  decimal  place,  the  product  should 
have  as  many  decimar  places  as  both  factors. 

If  the  number  of  figures  in  the  product  is  less  than  the  number  of  decimal 
places  in  the  two  factors,  supply  the  deficiency  by  prefixing  ciphers. 


RULE.  —  Multiply  as  in  integers,  and  from  the 
right  point  off  as  many  decimal  places  in  the  product  as 
there  are  decimal  places  in  the  two  factors. 

NOTE.  —  To  multiply  a  decimal  by  10,  100,  1000,  etc.,  remove  the  decimal 
point  as  many  places  to  the  right  as  there  are  ciphers  in  the  multiplier, 
annexing  ciphers  to  the  multiplicand,  11  necessary. 

EXAM  PLES. 

/      268.     1.  Multiply  three  hundred  forty-four  ten-thousandths 
/  by  twelve  thousandths. 

2.  Multiply  one  hundred  ninety-two  thousandths  by  four,  and 
\  nineteen  hundredths. 
V      ^What  is  six  hundredths  of  six  hundred  five  millionths  ? 


4.  What  is  five  hundredths  of  $864.32  ?     Of  $3645.75  ? 
\5.  What  is  .058£  of  784.65  ?     Of  943.25  ? 

6.  What  is  .99  x  1.106  x  .25  ?    4.105  x  .625  x  .512  ? 

Multiply  fr;Multiply 

f7.  8.716  by  .39  ;  by  .047.  jl/&   17.28  by  ^16f  ;  by  2.55f 

8.  .00865  by  .625  ;  by  97.75.       J13.  64.325  by  1.44f  ;  by  .06J. 

9.  .00128  by  8756.8;  by  7.865.   f-^.  86.75  by  1.33$;  by  5.76£. 

10.  387.25  by  .0147$;  by  .087-|.,J  15.  5.78  by  .0885;  by  .66f. 
\1.  58^625  by  .488}  ;  by  .375.      \  16.  237.5  by  .345$  ;  by  4.468$. 

17.  Multiply  1728  by  "v33£  ;  by  T25  ;  by  .125  ;  by  .20. 

18.  Multiply  .01837  by  1000  ;  .00145  by  100000  ;  .6874  by  100  ; 
5.375  by  10  ;  17.056  by  10000.     Find  the  sum  of  the  products. 


86  DECIMALS.  [Art.  269. 


DIVISION     OF     DECIMALS. 
269.     Ex.    Divide  .01728  by  1.44. 


OPERATION.  ANALYSIS.  —  Dividing  as  in  integers,  with- 

1.44  )  .01728  (  .012         out  reference  to  the  decimal  points  and  pre- 
\^  fixed  ciphers,  the  quotient  is  12.     Since  the 

dividend  is  the  product  of  the  divisor  and  quo- 
tient, it  must  contain  as  many  decimal  places 
288  .         as    both  of    them.      Hence    the    number  of 

decimal   places  in    the   quotient   must   equal 
the  number  in  the  dividend  less  the  number  in  the  divisor. 

SECOND  OPERATION.  ANALYSIS.  —  Since   multiplying  both  divisor  and 

.012         dividend  by  the  same  number  does  not  affect  the 

..    .  .  ->    O1  '79S         quotient,  make  the  divisor  a  whole  number  by  plac- 

ing the  point  two  places  to  the  right  (or  imagine  the 

point  to  be  omitted),  and  place  the  point  of  the  divi- 

288         dend  the  same  number  of  places  to  the  right.     (Com- 

2gg         pare  3rd  analysis,  page  72).     Indicate  the  new  posi- 

tion of  the  point  of  the  dividend  by  placing  an  index 

(  '  )  or  .pointer  between  the  figures  as  in  tjje  operation. 

1728  thousandths  divided  by  144  is  12  thousandths.  Observe  that  the  number 
of  decimal  places  in  the  quotient  is  equal  to  the  number  in  the  dividend  at 
the  right  of  the  pointer.  In  practice,  place  the  point  in  the  quotient  when 
all  the  figures  in  the  dividend  at  the  left  of  the  pointer  have  been  divided. 
Notice  in  the  operation  that  the  point  of  the  quotient  is  directly  above  the 
pointer  of  the  dividend,  and  that  each  figure  of  the  quotient  is  directly  above 
the  figure  of  the  dividend  which  produced  it. 


RULE.  —  Divide  as  in  integers,  and  point  off  from 
the  right  of  the  quotient  as  many  decimal  places  as  the  num* 
~ber  in  the  dividend  exceeds  the  number  in  the  divisor.  Or, 

Make  the  divisor  a  whole  number,  by  placing  the  point 
to  the  right,  and  place  the  point  of  the  dividend  the  same 
number  of  decimal  places  to  the  right.  Divide  as  in  inte- 
gers, and  place  a  decimal  point  in  the  quotient  when  the 
figures  of  the  dividend  have  been  used  as  far  as  the  new 
position  of  the  point  in  the  dividend. 

NOTES.  _  1.  If  the  number  of  figures  in  the  quotient  is  less  than  the  number 
of  decimal  places  to  be  pointed  off,  supply  the  deficiency  by  prefixing  ciphers. 

2.  If  the  divisor  contains  more  decimal  places  than  the  dividend,  before 
dividing  make  them  equal  by  annexing  ciphers  to  the  dividend.  If  necessary 
to  continue  the  division,  more  ciphers  may  be  added. 


Art.  270.]  DIVISION     OF    DECIMALS.  87 

3.  If,  after  dividing  all  the  figures  of  the  dividend,  there  is  a  remainder, 
the  division  may  be  continued  by  annexing  ciphers  (247,  1).    The  ciphers 
thus  annexed  must  be  regarded  as  decimal  places  of  the  dividend. 

4.  To  divide  a  decimal  by  10,  100,  1000,  etc.,  remove  the  decimal  point  as 
many  places  to  the  left  as  there  are  ciphers  in  the  divisor,  prefixing  ciphers  to 
the  dividend,  if  necessary. 

EXAMPLES. 

271.  1.  Divide  three  thousand  four  hundred  fifty-six  hun- 
dred-thousandths by  seventy-two  hundredths. 

2.  Divide  six,  and  twenty-five  hundredths  by  twenty-five  thou- 
sandths. 

Divide  Divide 

3.  35.88  by  .345  ;  by  4.16.  8.  .0648  by  .00425  ;  by  .0288. 

4.  .89958  by  .47  ;  by  .319.  9.  .31752  by  .648  ;  by  .00384. 

5.  12.6  by  14.4;  by-. 125.  10.  .1898  by  .33^;  by  .0048|. 

6.  96.3  by  .20  ;  by  .25.  11.  85.2451  by  4.56£;  by  8.27J. 

7.  5.27  by  1.24  ;  by  .85.  12.  45.367  by  .016| ;  by  l.OSOf 

13.  Divide  17.28  by  .20  ;  by  .25  ;  by  .33^  ;  by  .125  ;  by  .66f . 

14.  321  is  .178J  of  what  number  ? 

15.  186  is  five  hundredths  of  what  number  ? 

16.  What  must  37.375  be  multiplied  by  to  produce  448.5  ? 

17.  What  must  631.25  be  divided  by  to  produce  250  ? 

18.  Divide  176.824  by  100;  876.35  by  1000;  17380. 5  by  10000; 
2886.57  by  10  ;  375  by  1000000.     Find  the  sum  of  the  quotients. 

NOTE. — To  produce  a  result  in  hundredths  or  cents,  the  dividend  must 
contain  two  decimal  places  more  than  the  divisor  ;  to  produce  thousandths, 
three  places,  etc. 

Find  the  results  of  the  following  examples  in  hundredths,  and 
reduce  the  fractional  remainders,  if  any,  to  their  lowest  terms. 

Divide 

19.  $12. 52  by  $375.60;  by  $100.16. 

20.  $288  by  $1728  ;  by  $720. 

21.  $232.50  by  $3720  ;  by  $3875. 

22.  $60.40  by  $2416  ;  by  $1812. 

23.  $72  by  $3456  ;  by  $9000. 
84.  $212  by  $1484  ;  by  $508.80. 


88  DECIMALS.  [Art.  211. 

NOTE. — When  the  quotient  is  dollars  and  cents,  it  is  customary  in  busi- 
ness operations  to  omit  the  fraction  if  less  than  £  cent,  and  add  1  to  the  cents 
if  the  fraction  is  more  than  ^  cent. 

In  the  following  examples,  the  results  are  carried  to  cents  only  and  the 
factions  are  omitted. 

3       Divide  Divide 

26.  $18.08  by  .05  ;  by  .04.  28.  $720  by  .03^  ;  by  .07. 

26.  $648  by  .06  ;  by  13.  '•  29.  $12.25  by  .06| ;  by  .08. 

27.  $17.28  by  .48  ;  by  21.  80.  $960  by  .OOf  ;  by  .27. 

Divide  approximately  to  thousandths 

f  81.  176.4  by  13  ;  by  .17.   f  34.  120.96  by  70  ;  by  64. 

82.  229.48  by  50.72  ;  by  57,  35.  348.50  by  36  ;  by  .84. 

88.  91.20  by  65  ;  by  14.60.  f*  86.   1728  by  12.16  ;  by  17.5.  / 

212.  To  find  the  value  of  goods  sold  by  the  hun- 
dred or  thousand. 

\E&.    Find  the  cost  of  864  pounds  of  meal  at  $1.15  per 
dreq^ppunds  (civt.). 

OPERA1TON. 

8.64 

^  ^5  ANALYSIS. — 864  pounds   equal  8  hundred   weight  and 

.64  of  a  hundred  weight.     Hence  the  cost  of  864  pounds 

equals  8.64  times  $1.15,  or  $9.94. 
_9504_  C.  is  the  sign  for  hundred,  and  M.  for  thousand  (1C). 

$9.9360 

273.  RULE. — Reduce  the  quantity  to  hundreds  by  point- 
ing two  places  at  the  right,  or  to  thousands  by  pointing  off 
three  places.    Multiply  the  price  by  this  result  and  point  off 
the  product  as  in  multiplication  of  decimals. 

NOTE. — If  preferable,  multiply  the  price  by  the  quantity  as  given,  and 
point  off  two  additional  places  if  the  price  is  per  hundred,  and  three  additional 
places  if  the  price  is  per  thousand. 

EXAM  PLES. 

274.  1.  Find  the  cost  of  500  pounds  of  feed  at  $1.20  per  cwt. 

2.  Find  the  cost  of  4000  feet  of  boards  at  $8.50  per  thou- 
sand feet. 

3.  Find  the  cost  of  4375  feet  of  lumber  at  $11  per  thousand. 

4.  What  is  the  cost  of  13280  bricks  at  $7  per  thousand  ? 


Art.  274.]  REVIEW    EXAMPLES.  89 

5.  Find  the  cost  of  6500  cigars  at  $48  per  M. 

6.  What  is  the  value  of  640  pounds  of  hay  at  85c.  per  cwt.  ? 

7.  Find  the  freight  on   18480  pounds  of  merchandise  at  62 
cents,  per  civt. 

/°-  What  is  the  cost  of  5967  Ibs.  meal  at  $1.10  per  cwt.,  and 
4880  Ibs.  bran  at  75c.  per  cwt.? 

9.  Find  the  cost  of  26728  Ibs.  of  feed  at  $1.05  per  cwt. 

10.  Find  the  cost  of  11760  feet  joists  at  $14  per  thousand. 

11.  Find  the  cost  of  2  civt.  of  oatmeal  at  $2.40  per  cwt.,  and 
3  cwt.  of  cracked  wheat  at  $3.84  per  cwt. 

12.  What  is  the  cost  of  4  C.  bolts  at  $2.70,  and  J  0.  bolts  at 
$3.20  ? 

13.  Find  the  cost  of  insuring  a  house  for  $4500  at  35c.  per  $100. 

14.  Find  the  cost  of  25  M.  needles  at  $1.55  per  M. 

15.  Find   the   value   of   13450   feet   of   scantling  at   $18  per 
thousand. 

f      16.  What  is  the  cost  of  7J  M.  envelopes  at  $2.20  per  M.  ?  r 
V   17.  Find  the  cost  of  12400  sjiingles  at  $16  per  thousand 

V*^ 

REVIEW     EXAMPLES. 

275.     1.  Add  16  hundredths,   137  millionths,   48  ten-thou- 
sandths, and  2016  ten-millionths. 

2.  Add  16.07,  240.127|,  6.044,  27.1234. 

3.  Eeduce  ff-  to  a  decimal.         9.  Change  .8375  to  a  fraction. 

4.  From  175  take  16.083J.       10.  Multiply  117. 084- by  7.37|. 
6.  What  is  f  of  $175.75  ?         11.  Eeduce  .083$  to  a  fraction. 

6.  What  is  .33  of  187.5  ?          12.  From  375. If  take  198.88$. 

7.  Divide  43.75  by  .0125.         18.  1.75  is  £  of  what  number  ? 

8.  Divide  .06|  by  1.66|.  U-  What  is  .33$  times  1728  ? 

15.  $3.75  is  how  many  hundredths  times  $75  ? 

16.  $86.40  is  how  many  hundredths  of  $2592  ? 

17.  16.56  is  .05  of  what  number  ? 

18.  What  will  17280  bricks  cost  at  $3.25  per  M.  ? 

19.  If  278  barrels  of  pork  cost  $4378.50,  what  is  the  cost  of 
100  barrels  ? 

20.  Find  the  cost  of  12456  feet  of  plank  at  $8.75  per  M. 

21.  What  is  the  value  of  5  bbls.  sugar,  containing  312,  304, 
301,  305,  304  pounds  respectively,  at  9|  cents  per  pound  ? 


90  DECIMALS.  [Art.  275. 

22.  Find  the  cost  of  13f  pounds  of  crackers  at  15  J  cents  per 
pound. 

ANALYSIS. — Instead  of  multiplying  by  the  methofl  given  in  Art.  226, 
change  one  of  the  fractions  to  a  decimal  and  then  multiply.  Thus,  $.15£  = 
$.155.  13f  x  |.155  =  $3.131},  or  $2.13. 

Multiply  according  to  the  above  method 

33.  $3.17J  by  13} ;  by  19|,  28.  $4.12}  by  26} ;  by  llf 

24.  $.68fby  24J;  by  16}.  27.  $1.79£  by  37f;  by  44J. 

25.  $.88£  by  32£ ;  by  36J.  28.  $1.37£  by  18}  ;  by  45  J. 
29.  A    merchant    paid    for     merchandise    during    the    year 

$137618.75,  and  sold  merchandise  to  the  amount  of  $146347.87. 
What  was  the  gain,  if  the  net  market  value  of  the  merchandise 
remaining  unsold  was  $24378  ? 

80.  A  quartermaster  has  $8345  on  hand,  and  receives  $4379.62 
from  each  of  six  sales  of  property ;  he  turns  over  to.  quarter- 
master A  $2875.28,  and  pays  $120  for  corn.  Upon  being  relieved 
from  duty,  he  turns  over  to  quartermaster  B  one-third  of  the 
residue,  and  divides  the  remainder  equally  among  three  others, 
C,  D,  and  E.  What  was  paid  over  to  each  ? 

31.  Merchandise  on  hand,  Jan.  1,  1879,  $46312.85  ;  merchan- 
dise sold  during  the  year,  $317829.32  ;  merchandise  purchased  in 
the  same  time,  $301449.72  ;  merchandise  on  hand,  Dec.  31,  1879, 
$61378.12.  What  was  the  net  gain  or  loss  ? 

82.  A  farmer  sold  land  for  $22.50  an  acre,  as  follows  :  to  A, 
98|  acres ;  to  B,  f  of  the  number  sold  to  A ;  and  to  0,  J  the 
number  sold  to  A  and  B  both.  How  much  land  was  sold,  how 
much  did  B  and  0  each  receive,  and  what  was  the  amount  realized  ? 

33.  At  $28. 75  per  thousand,  how  many  feet  of  lumber  should 
be  given  for  2816  pounds  of  sugar  at  7y\  cts.  per  pound  ? 

84.  A  man  bequeaths  J  of  his  property  to  his  wife,  }  to  his 
son,  -j-  to  his  daughter,  and  the  remainder,  which  is  $36375,   to 
charitable  institutions.     What  is  the,  amount  bequeathed  to  each, 
and  the  total  amount  ? 

85.  A  gentleman  after  spending  -J-  of  all  his  money,  and  }  of 
the  remainder,,  had  $177.50  remaining  ;  how  much  had  he  at  first? 

36.  A  merchant  bought  100  yards  of  cloth  at  $3. 62 J  per  yard, 
and  87£  yards  at  $4.12J  per  yard.     At  what  average  price  per  yard 
should  he  sell  the  whole,  to  realize  a  profit  equal  to  -J-  of  the  cost  ? 

37.  If  31}  bushels  of  corn  cost  $17.50,  how  many  bushels  can 
be  bought  for  $616  ? 


DENOMINATE     NUMBERS. 


276.  A  Denominate  Number  is  a  concrete  number  (145), 
and  may  be  either  simple  or  compound. 

Denominate  numbers  are  used  to  express  divisions  of  time,  weights,  meas- 
ures, and  moneys  of  different  countries. 

The  scale  of  integers  and  decimals  is  uniform  ;  that  of  most  denominate 
numbers  is  varying. 

The  moneys  of  nearly  all  countries  excepting  Great  Britain,  and  the 
metric  system  of  weights  and  measures  have  a  uniform  decimal  scale. 

277.  A  Simple  Denominate  Number  refers  to  units  of 
the  same  name  and  value  ;  as  7  inches,  4  pounds. 

278.  A  Compound  Denominate  Number  refers  to  units 
of  different  names,  but  of  the  same  nature  ;  as  3  feet  6  inches,  4 
pounds  8  ounces. 


REDUCTION     OF     DENOMINATE 
NUMBERS. 

279.  Reduction  of  Denominate  Numbers  is  the  process 
of  changing  their  denomination  without  changing  their  value. 

28O.   To  reduce  denominate  numbers  from  higher  to 
lower  denominations. 

Ex.    How  many  pence  in  £8  16s.  Id.  ? 

OPERATION. 

£      s.       d.  ANALYSIS. — Since  there  are  twenty  shillings  in  1  pound, 

8     16      ?         in  8  pounds  there  are  8  times  20  shillings,  or  160  shillings. 
20  (For  convenience  multiply  by  20  as  an  abstract  number.) 

TTT  160  shillings  plus  16  shillings  equal  176  shillings.    Since 

there  are  12  pence  in  1  shilling,  in  176  shillings  there  are 
176  times   12  pence,  or  2112  pence.     2112  pence  plus  7 
176s.  pence  equal  2119   pence.     When  possible,  add  mentally 

..  -  the  number  of  the  lower  denomination  to  the  product,  as 

in  the  last  step  of  this  operation. 


2119^. 


92  DENOMINATE     NUMBERS.  [Art.  881. 

281.  EULE. — Multiply  the  number  of  the  highest  denom- 
ination given  by  the  nuiriber  of  the  next  lower  denomina- 
tion required  to  make  1  of  this  higher,  and  to  the  product 
add  the  given  number,  if  any,  of  such  lower  denomination. 

Treat  this  result,  and  the  successive  results  obtained,  in 
like  manner  until  the  number  is  reduced  to  the  required 
denomination. 

EXAMPLES. 

282.  Reduce  Reduce 

1.  £9  13s.  Wd.  to  pence.  13.  8  bu.  3  pk.  6  qt.  to  pints. 

2.  6  gal.  3  qt.  1  pt.  to  gills.  14.   13  gal.  3  qt.  1  pt.  to  pints. 

3.  £112  18s.  5d.  to  farthings.  15.  1  mi.  32  rd.  10  ft.  to  inches. 

4.  6  T.  12  cwt.  65  Ib.  to  pounds.  16.  29  sq.  rd.  to  square  feet. 

5.  8  mo.  16  da.  to  days.  17.  97  s<^.  rrf.  to  square  yards. 

6.  £75  17s.  7d.  to  pence.  7£.  5  sq.  mi.  to  acres. 

7.  £24515s.  3>r.  to  farthings.  75.  11  mo.  24  <fa.  to  days. 

8.  48  #w.  3^&.  6  qt.  to  quarts.  #0.   16  cords  112  CM.  ft.  to  cu.  ft. 

9.  7  T7.  9  cwt.  4:8  Ib.  to  pounds.  21.  £178  13s.  9rf.  to  pence. 
70.  18  $.  8  02.  to  pennyweights.  22.  2  ?/;•.  8  mo.  22  «?«.  to  days. 
11.  5  ww.  36  rd.  lift,  to  feet.  ££.   13  T.  17  cwtf.  82  Ib.  to  pounds. 
70.  456  miles  to  feet.  24.  £31  11s.  lid.  to  pence. 

283.  To  reduce  denominate  numbers  from  lower  to 
higher  denominations. 

Ex.    Reduce  2119  pence  to  higher  denominations. 

OPEBATION.  ANALYSIS. — Since   there  are  12  pence  in   1 

12  )  2119^.  shilling,  in  2119  pence  there  are  as  many  shillings 

9oTT7fi     _L  7  7  as  12  pence  are  contained  times  in  2119  pence,  or 

176  shillings,   and   7  pence   remaining.       Since 
£8     -j-  16s.  there  are  20  shillings  in  1  pound,  in  176  shillings 

£g  ^g      v^        there  are  as  many  pounds  as  20  shillings  are  con- 
tained times  in  176  shillings,  or  8  pounds,  and  16 
shillings  remaining.     Therefore,  2119&  =  £8  16s.  Id. 

284:.  RULE. — Divide  the  given  number  ~by  the  number  of 
that  denomination  required  to  make  1  of  the  next  higher, 
reserving  the  remainder,  if  any,  as  part  of  the  answer. 

Treat  the  quotient,  and  the  successive  quotients  obtained, 
in  like  manner  until  the  number  is  reduced  to  the  required 
denomination.  The  last  quotient  and  the  several  remain- 
ders will  form  the  answer. 


Art.  885.]  REDUCTION  OF  DENOMINATE  FRACTIONS.        93 

EXAMPLES. 

285.  Reduce  Reduce 

1.  8475d.  to  pounds.  11.  13387^.  to  pounds. 

2.  9683  cu.fi.  to  cords.  12.  987/2.  to  yards. 

3.  7534  pts.  to  bushels.  13.  17416  Ibs.  to  tons. 

4.  ^11  pts.  to  gallons.  14.  1809  m.  to  yards. 

5.  987  m.  to  yards.  15.  4711  pts.  to  bushels. 

6.  216  da.  to  months.  16.  8370d.  to  pounds. 

7.  875  rods  to  miles.  17.  5316  sq.  rds.  to  acres. 

8.  6375  Ars.  to  weeks.  18.  7418  o«.  to  cw#. 

9.  9537  sec.  to  hours.  19.  8716  CM.  /*.  to  cords. 
10.  6239  w.  to  yards.  20.  4829d.  to  pounds. 

REDUCTION     OF    DENOMINATE 
FRACTIONS. 

286.  A  Denominate  Fraction  is  a  fraction  whose  integral 
unit  is  a  denominate  number. 

The  principles,  analyses,  and  rules  of  denominate  fractions  are  essentially 
the  same  as  those  of  denominate  integers  ;  therefore,  no  special  rules  are 
necessary  for  their  reduction.  A  sufficient  number  of  illustrative  examples 
are  given  to  fully  explain  the  different  cases  that  may  arise. 

287.  To  reduce  denominate  fractions  from  higher  to 
lower  denominations. 

Ex.    Reduce  T\  of  a  £  to  pence. 

OPEBATIONS.  ANALYSIS.—  Since  there  are  20  shillings 

5  in  £1,  in  TV  of  a  £  there  are  T7^  of  20  shil- 

A  X  -^  :  :  ¥*•  lings,  or  ^  shillings.     Since  there  are  12 

3  pence  in  1  shilling,  in  *£•  shillings  there 

^   X  ^  =  105d.  are  ^  times  12  pence,  or  105  pence.    Or, 

multiply  the  given  fraction  by  the  num- 


r»      _i_       »BI        A  __  i  n^/7         ^ers  °^  ^e  sca^e  re(lllired  to  reduce  its  de- 
'  V  >  nomination  to  the  required  denomination. 

Ex.    Reduce  ^  of  a  £  to  shillings  and  pence. 

ATI6°N'  ANALYSIS.—  Multiplying  by  20,  £^  =  8f 

^  X  ^  —  3/  =  8J5.  shillings.    Reserve  the  integral  part  of  the 

3  result,  and  reduce  the  fractional  part  to  pence. 

I    X  \*  =  9d.  Multiplying   by    12,    f    shilling  =  9  pence. 

Hence'  *A  -  ^  M- 


94  DENOMINATE     NUMBERS.  [Art.  288. 

EXAMPLES. 

288.  1.  Beduce  -^  of  a  £  to  pence. 

2.  Keduce  ^  of  a  £  to  integers- of  lower  denominations. 

8.  Keduce  f  of  a  mile  to  feet. 

4.  Change  -f^  of  a  ton  to  pounds. 

5.  Reduce  f  J  of  a  £  to  shillings  and  pence. 

6.  Change  -J-f  of  a  bushel  to  pints. 

7.  Eeduce  ff  of  a  mile  to  feet. 

8.  Reduce  -J  of  a  hundred- weight  to  integers  of  lower  denom- 
inations. 

9.  Reduce  ^-J-  of  a  gallon  to  pints. 

10.  Change  |f  of  a  cord  to  cubic  feet. 

289.  To  reduce  denominate  decimals  from  higher  to 
lower  denominations. 

Ex.    Reduce  .4375  of  a  £  to  pence. 

ANALYSIS. — Since  there  are  20  shillings  in    £1,    in 
£.4375,  there  are  .4375  x  20  shillings,  or  8.75  shillings. 
8.7500.5.        Since  there  are  12  pence  in  1  shilling,  in  8.75  shillings, 
12  there  are  8.75  x  12  pence,  or  105  pence. 


OPERATION. 

£.4375 
20 


105.0000^. 

Ex.    Reduce  .4375  of  a  £  to  integers  of  lower  denominations. 


OPERATION. 


ANALYSIS.— Multiplying   by  20,  £.4375  =  8.75    shil- 
lings.   Reserve  the  integral  part  of  the  result,  and  reduce 
S.  8J.7500        the  decimal  part  to  pence.   Multiplying  by  12,  .75  shilling 
12         =9  pence.    Hence,  £.4375  =  Ss.  Qd. 

d.  9|.0000 

EXAMPLES. 

29O.     1.  Reduce  .625  of  a  £  to  pence. 

2.  Reduce  .875  of  a  £  to  shillings  and  pence. 

8.  Reduce  .6375  of  a  £  to  pence. 

4.  Reduce  .6825  of  a  £  to  shillings  and  pence. 

5.  Change  2.333J  yrs.  to  integers  of  lower  denominations. 

6.  Change  £16.467  to  integers  of  lower  denominations. 


Avt.  290.]  REDUCTION  OF  DENOMINATE  FRACTIONS.       95 

7.  If  1  pound  sterling  can  be  bought  for  $4.87,  how  many 
pounds  can  be  bought  for  $1000  ? 

8.  Reduce  2.417  yr.  to  integers  of  lower  denominations. 

9.  Reduce  £15.3375  to  integers  of  lower  denominations. 

10.  A  certain  sum  at  a  certain  rate  will  in  1  yr.  produce  $60 
interest ;  in  what  time  will  the  same  sum  at  the  same  rate  produce 
$15.50  interest  ? 

291.  To   reduce   denominate   numbers   to    fractions 
of  higher  denominations. 

Ex.    Reduce  f  of  a  penny  to  the  fraction  of  a  £. 

OPERATIONS.  ANALYSIS. — Divide  the  given  fraction 

A   __:_  12  =    Jn-  S.  ^°J  the  numbers  of  the  scale  required  to  re- 

i     .    o  A  i     f  duce  pence  to  pounds. 

If  the  answer  is  required  in  the  form  of 

/-\      ;a         ivi    i-fi        a  decimal,  reduce  the  resulting  fraction  to  a 

<  Y*  X  inr  -  decimal  by  Art.  254.    £^  =  £.0025. 

Ex.    Change  9  pence  to  a  fraction  of  a  £. 

OPERATIONS.  ANALYSIS. — For  first  operation,  as  in  pre- 

I  X  ^  X  ,V  =  iro^-         Vious  examPle- 

4  Or,  since  there   are  240  pence   in  £1,  1 

Or,   £r^  =  £gV  penny  equals  ^  of  a  £,  and  9  pence  equal 

T&S,  or  &  of  a  £. 

Ex.    Reduce  125.  9d.  to  the  fraction  of  a  £. 

OPERATION. 

12s.  9d.  =  153^.  ANALYSIS. — 12  shillings  9  pence  =  153  pence. 

£^  __  240d        Since  £1  =  240  pence,  1  penny  equals  ^  of  a  £, 

and  153  pence  equal  £ff,  or  f  £  of  a  £. 
iff  —  ft*' 

EXAMPLES. 

292.  1.  Reduce  f  of  a  penny  to  the  fraction  of  a  pound. 

2.  Reduce  420  grains  to  the  fraction  of  an  ounce  Troy. 

3.  Change  275  feet  to  the  fraction  of  a  mile. 

4.  Reduce  49£  feet  to  the  fraction  of  a  mile. 

5.  Reduce  165.  lOd.  to  the  fraction  of  a  pound. 

6.  Reduce  3s.  6d.  to  the  fraction  of  a  pound. 

7.  Reduce  6  mo.  20  da.  to  the  fraction  of  a  year. 

8.  Reduce  84  pounds  to  the  fraction  of  long  ton  (2240  pounds). 

9.  Reduce  3  qt.  1  pt.  to  the  fraction  of  a  bushel. 
10.  Reduce  16s.  8d.  to  the  fraction  of  a  pound. 


96  DENOMINATE     NUMBERS.  [Art.  293. 

293.   To  reduce  denominate  numbers  to  decimals  of 
higher  denominations. 

Ex.    Reduce  .  6  of  a  penny  to  the  decimal  of  a  £. 

OPERATION.  ANALYSIS.—  Divide  the  given  decimal  by  the  num- 

12  )  .6    d.          bers  of  the  scale  required  to  reduce  pence  to  pounds. 
20  ^    05  <?  ^  ^ne  answer  i§  required  in  the  form  of  a  fraction, 

reduce  the  resulting  decimal  to  a  fraction  by  Art.  258. 


£.0025 

Ex.    Reduce  9  pence  to  the  decimal  of  a  £. 

OPERATIONS.  ANALYSIS.—  For  first  operation,  as  in 

12  )  9.  d.  previous  example. 

OQ  \    75  Or,  since  there  are  240  pence  in  £1, 

1  penny  equals  -fa  of  a  £,  and  9  pence 

.0375  £.  equal  ¥|7,  or  /„  of  a  £.    £^  =  £.0375 

Or,  £^  =  £*V  =  £.0375.       <254): 

Ex.    Reduce  £18  12s.  Sd.  to  the  decimal  of  a  £. 

OPERATION.  ANALYSIS.  —  Write  the  denominations  given  in  a 

12  )     9.       d.         vertical  column,  the  lowest  denomination  at  the  top. 

20  \  12  75  e  Since  there  are  12  pence  in  1  shilling,  9  pence  are  equal 

to  .75  shilling  ;  to  which  annexing  the  12  shillings 

£18.6375         given  we  have  12.75  shillings.     Since  there  are  20  shil- 

lings  in  £1,  12.75  shillings  are  equal  to  £.6375,  to 

which  annexing  the  £18,  we  have  £18.6375.     Hence  £18  12s.  Qd.  —  £18.63750 

EXAM  PLES. 

294.     1.  Reduce  .  875  of  a  shilling  to  pounds. 
2.  Change  12  cwt.  to  the  decimal  of  a  ton. 
8.  What  decimal  of  a  £  are  18s.  6d.  ? 

4.  Reduce  £14  15s.  9^.  to  the  decimal  of  a  pound. 

5.  Reduce  116  cu.  ft.  to  the  decimal  of  a  cord. 

6.  Reduce  £247  14s.  3d.  to  pounds. 

7.  What  decimal  of  an  acre  are  64  sq.  rds.  ? 

8.  Reduce  £27  10s.  6d.  to  pounds. 

9.  What  is  the  cost  of  16  tons  12  cwt.  of  "Nut"  coal  at  $6.80 
per  ton,  and  8  tons  16  cwt.  of  "  Chestnut  "  coal  at  $6.10  per  ton  ? 

10.  If   1  pound  is  equivalent  to  I4.87J-,  what  is  the  value  of 
£123  16s.  9d.  in  IT.  S.  money  ? 

11.  Reduce  £25  12s.  6d.  to  the  decimal  of  a  £,  multiply  the 
result  by  .03,  and  reduce  the  resulting  decimal  to  shillings  and 
pence. 


Art.  295.]    ADDITION  OF  DENOMINATE  NUMBERS.  97 


ADDITION   OF   DENOMINATE    NUMBERS. 

295.  Denominate  numbers  are  added,  subtracted,  multiplied, 
and  divided  by  the  same  general  methods  as  are  employed  for  like 
operations  in  abstract  numbers.  The  only  difference  arises  from 
the  use  of  a  varying  scale  instead  of  the  uniform  scale  of  10. 

•      Ex.    Add  £5  11s.  4d.,  £7  145.  9d.,  £6  16s.  Sd.,  and  £7  5s.  9d. 
OPERATION.  ANALYSIS. — Write  the  numbers  so  that  like  denomina- 

£  s.  d.  tions  stand  in  the  same  column,  and  begin  to  add  at  the  right. 
5  11  4  The  sum  of  the  pence  is  30d.  =  2s.  6d.  Write  the  Qd.  under 
7  14  9  ^ne  c°lumn  of  pence,  and  add  the  2s.  to  the  column  of  shil- 
g  -^g  g  lings,  obtaining  for  the  sum  48s.  =  £2  8s.  Write  the  8s. 
under  the  column  of  shillings,  and  add  the  £2  to  the  column 
*  °  of  pounds,  obtaining  for  the  sum  £27  ;  which  write  under 


27        86      the  column  of  pounds,  producing  the  entire  sum,  £27  8s.  6d. 

EXAMPLES. 

296.  1.  Add  £16  5s.  4d.,  £12  85.  9d.,  £13  14s.  Sd.,  £42  Os.  7d., 
and  18s.  Gd. 

2.  Add  3  T.  19  cwt.  46  lb.,  4  T.  13  cwt.  14  lb.,  18  T.  13  cut, 
24  lb.,  and  42  T.  Scivt.  82  lb. 

3.  Add  £163  16s.  lid.,  £52  8s.  Qd.,  £3  14s.  2d.,  £84  12s.  lid., 
£106  Is.  4d.,  and  £49  13s.  Sd. 

4.  Add  1  yr.  6  mo.  10  da.,  3  yr.  8  mo.  24  da.,  4  yr.  11  mo.  16  da., 
3  mo.  18  d«.,  and  1  yr.  8  mo.  8  da. 

5.  Add  8  cd.  106  cu.  ft.,  3  cd.  85  cu.  ft.,  and  2  cd.  113  cw.  /*. 

6.  Add  16  hr.  43  mm.  48  sec.,  3  7^r.  12  min.  40  sec.,  1  ^r.  49  min. 
13  sec.,  and  5  Ar.  19  sec. 

7.  Add  4  fa*.  3  ^.  6  qt.  1  ^.,  10  bu.  2  p&.  7  gtf.  Ipt.,  11  iw. 
3  jP&.  1  qt.  I  pt.,  9  #w.  2  pk.  5  <7#.  1  ^t?^. 

8.  7  yd.  2ft.  10  in.,  2  #d.  l/^.  9  in.,  8  yd.  1  /£.  8  in.,  6  yrf. 
4m.,  1  yd.  2ft.  9  m. 

9.  Add  1  $.  11  oz.  ISpwt.  14  ^r.,  2  lb.  8  02.  lOpwt.,  4  $.  5  02. 
18  ^/r.,  and  10  oz.  13  pwt.  12  gr. 

10.  Add  16  #aZ.  3  ^.  1  ^.,  45  gal.  2  qt.,  17  gal.  1  qt.  Ipt., 
I  gal.  3  gtf.,  15  gal.  I  pt.,  and  24  gal.  3  gtf.  1  jo^. 

^^.  Add  £17  16s.  Sd.,  £37  13s.  6d.,  £46  7d.,  £11  5s.  Wd.,  £8 
4s.,  £38  19s.  3d.,  and  £45  12s.  Sd. 

12.  Add  £175  14s.  9rf.,  £37  9s.  3d.,  £5  10s.  9d.,  17s.  3d.,  £55 
17s.,  £3  6s.  9d.,  £44  18s.  5d.,  £318  15s.  6d.,  and  £3  11s.  lid. 


98  DENOMINATE     NUMBERS.  [Art.  297. 


SUBTRACTION     OF    DENOMINATE 
NUMBERS. 

297.  Ex.    From  £10  6s.  4d.  take  £8  15s.  3d. 

OPERATION.  ANALYSIS. — Write  the  numbers  so  that  like  denomina- 

£       s.       d.        tions  stand  in  the  same  column,  and  begin  to  subtract  at  the 

10        64        right.     M.  from  4d.  leaves  Id.,  which  write  under  the  col- 

8     15     3         umn  of  pence.     Since  15s.  cannot  be  subtracted  from  6s., 

~~I      7Z      7         take  £1  =  20s.  from  £10,  leaving  £9,  and  add  it  to  the  6s., 

making  26s.     15s.  from  26s.  leaves  11s.,  which  write  under 

the  column  of  shillings.    £8  from  £9  leaves  £1,  which  write  under  the  column 

of  pounds.     Hence  the  difference  required  is  £1  11s.  Id. 

EXAMPLES. 

298.  1.  From  £175  16s.  Sd.  take  £87  12s.  6d. 

2.  From  £84  10s.  2d.  take  £63  6s.  Wd. 

3.  From  £16  6s.  lid.  take  £12  12s.  Sd. 

4.  From  £48  10s.  Sd.  take  £24  16s.  lOd. 

5.  From  16  yr.  8  mo.  10  da.  subtract  12  yr.  5  mo.  8  da. 

6.  From  80  yr.  10  mo.  16  da.  take  76  yr.  5  mo.  24  da. 

7.  From  81  yr.  4  mo.  25  da.  take  80  yr.  10  mo.  15  da. 

8.  From  82  yr.  3  mo.  20  da.  take  79  yr.  8  mo.  26  da. 

MULTIPLICATION    OF    DENOMINATE 
NUMBERS. 

299.  Ex.    Multiply  £7  16*.  Sd.  by  11. 

OPERATION.  ANALYSIS. — 11  times  Sd.  are  SSd.  =  7s.  4.    Write  the 

£      s.      d.        4^,  under  the  pence,  and  add  the  7s.  to  the  product  of  shil- 

7     16     8        lings.     11  times  16s.  are  176s.,  plus  7s.  from  the  preceding 

11         product  are  183s.  =  £9  3s.    Write  the  3s.  under  the  shil- 

^        q 7         lings,  and  add  the  £9  to  the  product  of  pounds.     11  times 

£7  are  £77,  plus  £9  from  the  preceding  product  are  £86, 

which  write  under  the  pounds.    Hence  the  entire  product  is  £86  3s.  4d. 

Ex.    Multiply  £8  12*.  6d.  by  .05. 

ANALYSIS. — Reduce  the  multiplicand  to  the  decimal  of  a  pound  by  Art. 
293,  perform  the  required  multiplication,  and  reduce  the  result  to  shillings 
and  pence  by  Art.  289. 

£8  12s.  Qd.  =  £8.625.     £8.625  x  .05  =  £.43125.    £.43125  =  8s.  7.5d. 

Or,  reduce  the  multiplicand  to  pence  by  Art.  281,  perform  the  required 
multiplication,  and  reduce  the  result  to  shillings  and  pounds  by  Art.  284.  , 
£8  12s.  Qd.  =  2070d.  2070&  x  .05  =  103.5d.  103.5d  =  8s.  1M. 


Art.  30O.]     DIVISION  OF  DENOMINATE  NUMBERS.  99 


EXAM  PLES. 

,      3OO.     1.  Multiply  £17  10*.  Sd.  by  7  ;  by  9  ;  by  11 ;  by  15. 

2.  How  many  cords  of  wood  in  12  loads,  each  load  containing 
2  cd.  108  cu.  ft.  ? 

3.  What  is  the  cost  of  25  yd.  of  silk,  at  £1  2*.  6d.  per  yd.  ? 

4-  Find  the  weight  of  24  spoons,  each  spoon  weighing  1  oz. 
13  pwt. 

5.  Multiply  1  hr.  38  min.  22  sec.  by  10  ;  by  12  ;  by  15  ;  by  18. 

6.  If  15  men  perform  a  certain  piece  of  work  in  3  da.  1C  hr. 
52  min.,  how  long  would  it  take  one  man  to  perform  it  ? 

7.  What  will  50  gal.  of  wine  cost  at^  85.  3d.  per  gallon  ? 

8.  What  is  .05  of  £127  16s.  6d.  ?     Of  £145  15s.  9d.  ? 

9.  Multiply  £138  8*.  9d.  by  .02J ;  by  .04 ;  by  .06  ;  by  .07. 

DIVISION   OF    DENOMINATE   NUMBERS. 

3O1.     Ex.    If  6  yds.  of  cloth  are  worth  £8  185.  6d.,  what  is 
1  yd.  worth  ? 

OPERATION.  ANALYSIS. — 1  yd.  is  worth  1  stalA  as  much  as  6  yd. 

&      s.      d.        i   Of  £8  is  £i  an(j  £2  remaining.     Write  the  £1  in  the 

6  )  8     18 6        quotient,  and  reduce  the  £2  to  shillings.     £2=40s.,  plus 

^        99        18s.  in  the  dividend  =  58s.    -J-  of  58s.  is  9s.  and  4s.  re- 
maining.    Write  the  9s.  in  the  quotient,  and  reduce  the 
4s.  to  pence.    4s.  =48d.,  plus  &d.  in  the  dividend  =  54d.    £  of  54d.  is  9d., 
which  write  in  the  quotient.     £1  9s.  9d.  is  the  quotient  required. 

NOTE.— When  the  divisor  is  a  denominate  number,  as  in  Ex.  2,  reduce 
both  divisor  and  dividend  to  the  same  denomination,  and  divide  as  in  simple 

numbers. 

EXAM  PLES. 

3O2.     1.  Divide  £13  12s.  3d.  by  11 ;  by  9  ;  by  33. 

2.  How  many  yards  of  muslin  at  Id.  per  yard  can  be  bought 
for  £5  126-.  ?     For  £9  9*.  ?    For  £10  5s.  4d.  ?     (See  note. ) 

3.  Plow  many  yards  of  silk  at  £1  195.  2d.  per  yard  can  be  pur- 
chased for  £86  3s.  ±d. ?     (See  note.) 

4.  Divide  85°  18'  30"  by  12  ;  by  15  ;  by  18  ;  by  27. 

5.  Divide  322  A.  90  sq.  rd.  by  8  ;  by  10  ;  by  13  ;  by  16. 

6.  If  42  yd.  of  cloth  cost  £20  16,9.  6d.,  what  is  the  price  of 
1  yd.  ?    Of  12  yd.  ?     Of  20  yd.  ?    Of  37  yd.  ? 

7.  Divide  17*.  3d.  by  .02^  ;  by  .04  ;  by  .05  ;  by  .09  ;  by  .15. 
Reduce  the  dividend  to  the  decimal  of  a  pound  or  shilling,  divide  in  the 

oeual  manner,  and  reduce  the  quotient  to  pounds,  shillings,  and  pence. 


100  DENOMINATE     NUMBERS.  [Art.  3O3- 


DIVISIONS     OF    TIME. 

303.  The  natural  divisions  of  time  are  the  year  and  the  day, 
the  other  divisions  being  artificial. 

The  year  is  the  time  in  which  the  earth  makes  one  revolution  around  the 
sun.  The  day  is  the  time  in  which  the  earth  makes  one  revolution  on  its  axis. 

TABLE. 

60  Seconds  (sec.)  =  1  Minute      .     .     .     .     .     .  min. 

60  Minutes  =  1  Hour hr. 

24  Hours  =  1  Day da. 

7  Days  =  1  Week wk. 

365  Days,  ^ 

52  Weeks,  1  day,  or  >  =  1  Common  Year   ....  yr. 
12  Calendar  Months  J 

366  Days  =  1  Leap  Year yr. 

100  Years  —  1  Century '  C. 

NOTE. — In  many  business  transactions  the  year  is  regarded  as  360  days, 
or  12  months  of  30  days  each. 

304.  The  Solar  Day  is  the  interval  between  two  consecutive 
returns  of  the  sun  to  the  meridian. 

On  account  of  the  varying  motion  of  the  earth  around  the  sun,  the 
solar  days  are  of  unequal  length.  For  civil  purposes  in  measuring  time,  the 
average  of  all  the  days  in  the  year  is  taken  as  the  unit. 

305.  The  Solar  Year  is  the  time  between  two  consecutive 
returns  of  the  sun  to  the  vernal   equinox.     Its  exact  length  is 
365  da.  5  lir.  48  min.  50  sec.  in  mean  solar  time.     For  civil  pur- 
poses, the  year  consists  of  365  or  366  days. 

In  the  calendar  established  by  Julius  Csesar,  B.C.  46,  and  thence  called  the 
Julian  calendar,  three  successive  years  were  made  to  consist  of  365  days  each ; 
and  the  fourth,  of  366  days.  According  to  the  Julian  calendar,  the  average 
length  of  the  year  was  365|-  days,  thus  making  an  error  of  11  min.  10  sec.  each 
year  ;  which  in  400  years  would  amount  to  73  hours,  or  about  3  days.  In  the 
sixteenth  century,  in  consequence  of  the  excess  of  the  Julian  year  above  the 
.true  solar  year,  the  error  in  the  calendar  was  10  days.  To  correct  the  calen- 
dar, and  to  prevent  any  error  in  the  future,  Pope  Gregory  XIII.  decreed  that 
10  days  should  be  omitted  in  the  month  of  October,  1582,  and  that  all  centen- 
nial years  not  divisible  by  400  should  be  common  years.  This  calendar  is 
sometimes  called  the  Gregorian  calendar.  It  is  now  used  in  all  civilized 
countries  except  Russia. 


Art.  305.] 


DIVISIONS     OF 


101 


The  Julian  and  Gregorian  calendars  are  also 
Style  and  New  Style.  In  consequence  of  theaters  i^CO  and  1$$  being j .)op- 
mon  years  by  the  Gregorian  calendar,  the  difference  between  the  two  styles  is 
now  12  days.  Thus,  when  it  is  July  4  in  Russia,  it  is  July  16  in  other  countries. 

306.  RULE  FOR  LEAP  YEARS. — All  years  divisible  by  4> 
except  centennial  years,  are  leap  years. 

All  centennial  years  divisible  by  400  are  leap  years. 

307.  The  Calendar  Months,  with  the  number  of  days  they 
contain,  are  as  follows  : 


Season. 


WINTER. 


SPRING 


f  3.  Marc] 
NG.    •!  4.  April 
1 5.  Mav 


Days. 

January  (Jan.)  31. 
February  (Feb.)  28. 
in  leap  year  29. 
March  (Mar.)  31. 
(Apr.)  30. 

May  31. 


SUMMER. 


AUTUMN 


WINTER. 


C  9.  S 

JMN.   -!    10.    C 

111.  J 


Days. 

6.  June  30. 

7.  July  31. 

8.  August  (Aug.)       31. 

9.  September  (Sep.)  30. 
October  (Oct.)        31. 
November  (Nov.)  30. 

12.  December  (Dec.)   31. 


The  number  of  days  in  each  month  may  be  easily  remembered  from  the 
following  lines : 

"  Thirty  days  hath  September, 
April,  June,  and  November; 
February  twenty-eight  alone, 
All  the  rest  have  thirty-one ; 
Except  in  Leap  year,  then  is  the  time 
When  February  has  twenty-nine." 

EXAMPLES. 

3O8.     1.  Reduce  2  ivk.  4  da.  16  lir.  40  min.  to  minutes. 

2.  How  many  days  in  7  mo.  22  da.  ? 

3.  Reduce  2.375  years  to  years,  months,,  and  days. 

4'  If  a  person's  income  is  $1000  per  day,  how  much  is  that 
per  minute  ? 

5.  From  88  yr.  8  mo.  10  da.  subtract  86  yr.  5  mo.  24  da. 

6.  Multiply  3  hr.  24  mm.  32  sec.  by  15. 

7.  How  many  leap  years  from  1886  to  1897  ?     From  1795  to 
1827  ?    From  1887  to  1903  ? 

8.  How  long  from  14  min.  40  sec.  past  9  A.  M.  to  37  min. 
30  sec.  past  5  P.  M.  ? 

9.  Reduce  100000  sec.  to  higher  denominations. 

10.  Find  the  value  of   76£  hours  of  labor  at  $3.50  per  day  of 
8  hours. 


102  DXXbMINATE     NUMBERS.  [Art.  3O9. 

3O!>.  To  fip.d  the  interval  of  time  between  two  dates. 

31O.  There  are  two  raethods  in  common  use  for  finding  the 
time  between  two  dates  :  1,  by  compound  subtraction,  in  which 
the  result  is  given  in  years,  months,  and  days,  and  in  which  12 
months  are  considered  a  year,  and  30  days  a  month  ;  2,  the  result 
is  given  in  days,  or  in  years  and  days,  and  the  true  number  of 
days  is  taken  for  each  month. 

Ex.  Find  the  time  in  months  and  days  from  Apr.  24  to 
Nov.  10. 

OPERATION.  ANALYSIS.  —  Represent  the  months  and  days  by  their  num- 

mo.     da.        bers  and  find  their  difference  by  compound  subtraction  (297), 
11     10        writing  the  later  date  as  the  minuend  and  the  earlier  as  the 

4  24        subtrahend. 

In  many  examples  the  interval  may  be  found  mentally  as 

follows  :  From  Apr.  24  to  Oct.  24  are  6  mo.  ;  in  Oct.  there  are 

6  more  days  after  the  24th  (regarding  each  month  as  30  days),  and  in  Novem- 

ber to  Nov.  10th  inclusive,  there  are  10  days.     Hence  the  total  time  between 

the  given  dates  is  6  mo.  16  da. 

The  above  methods  may  be  used  for  finding  the  exact  interval  in  days  by 
making  the  necessary  corrections,  6  mo.  16  da.  =  196  da.  From  Apr.  24  to 
Nov.  10,  there  are  4  months  containing  31  da.  each  ;  hence  the  true  answer  is 
196  da.  +  4  da.,  or  200  da. 

NOTE.  —  When  the  month  of  February  is  included,  subtract  2  days  in  a 
common  year,  and  1  day  in  a  leap  year. 

Ex.    Find  the  time  from  May  18,  1884,  to  Mar.  2,  189G. 

OFBRATION. 

yr.   mo.    da. 

90     3       2 

ANALYSIS.  —  As  in  preceding  example. 
84     5     lo 

5  9     14 

Ex.  What  is  the  exact  number  of  days  from  July  20.  1888, 
to  Nov.  10,  1889  ? 

OPERATION.  ANALYSIS.  —  In  finding 

365  from  July  20,  1888,  to  July  20,  1889.     the  interval  between  two 

11  remaining  in  July.  dates  the  last  dar  is  count- 

31  in  August.  *•  and  not  the  fil^  Since 

the  time  is  more  than  one 

30  in  September.  year>  write  down  365  dayg 

31  in  October.  as  the  number  of  days  from 
10  in  November.                                               the  first  date  to  the  same 


m  from  July  *,,  1888,  to  Nov.  10,  1889. 

days  in  the  month  of  July  after  the  20th,  then  the  number  of  days  in  each 


Art.  31O.]  DIVISIONS     OF    TIME.  100 

of  the  full  calendar  months,  and  finally  the  number  of  days  in  November  to 
Nov.  10  inclusive.  The  sum  of  these  numbers  will  be  the  required  time. 

In  solving  examples  by  this  method,  the  student  should  remember  that  the 
first  quarter  contains  90  days,  the  second  quarter,  91  days,  the  third  quarter, 
92  days,  and  the  last  quarter,  92  days;  the  first  six  months,  181  days,  and  the 
last  six  months,  184  days. 

EXAMPLES. 

311.  Find  the  time  by  compound  subtraction  from 

1.  Jan.  10  to  Aug.  28. 

2.  Mar.  16  to  Dec.  4. 

3.  Feb.  5,  1886,  to  Oct.  16,  1887. 

4.  Jan.  27,  1885,  to  July  4,  1887. 

5.  May  16,  1886,  to  Mar.  24,  1887. 

6.  June  28,  1885,  to  Apr.  10,  1886. 

7.  July  30,  1886,  to  May  12,  1887. 

8.  Aug.  16,  1887,  to  Jan.  1,  1888. 

Find  also  the  exact  number  of  days  between  the  above  dates. 

9.  How  many  days  from  Jan.  1,  1888,  to  Jan.  1,  1906  ? 

10.  Ninety  days  after  June  21  is  what  date  ? 

OPERATIONS.               ,  ANALYSIS. — Subtract  from  the  given 

90                     Or,    9  June.  number  of  days,  the  number  of  days  re- 

9     June.             31  July.  maining  in  June,  and  from  this  remainder, 

31  Auff  subtract  successively  the  number  of  days 

in  the  following  months  until  the  remain- 

31     July.              71  der  is  equal  to  or  less  than  the  number  of 

5Q                            90  days  in  the  next  following  month.    The 

last  remainder  represents    the  required 
31     Aug.  19     Sept.         date_ 

19     Sept.  Or,  write  the  remaining  number  of 

days  in  June,  and  the  number  of  days 

in  a  sufficient  number  of  months  to  produce  about  the  given  number  of  days. 
Take  their  sum  and  subtract  it  (if  possible)  from  the  given  number  of  days. 
The  remainder  will  be  the  day  of  the  following  month  representing  the 
required  date.  If  the  sum  is  greater  than  the  given  number,  subtract  the 
excess  from  the  number  of  days  in  the  last  month  written.  The  remainder 
will  be  the  required  date. 

If  the  time  be  30,  60,  or  90  days,  regard  each  30  days  as  a  calendar  month, 
and  correct  by  subtracting  1  day  for  each  intervening  month  containing  31 
days,  and  adding  2  days  for  February  (in  leap  year  1  day).  Thus  3  months 
after  June  21  is  Sept.  21,  and  by  subtracting  2  days  for  July  and  August,  the 
correct  result  is  Sept.  19. 

11.  63  days  after  Oct.  4  is  what  date  ? 

12.  90  days  after  Mar.  24  is  what  date  ? 


104 


DENOMINATE     NUMBERS. 


[Art.  312. 


LINEAR     MEASURES. 


Linear  or  Long  Measure  is  used  in  measuring  dis- 
tances ;  also  the  length,  breadth,  and  height  of  bodies,  or  their 
linear  dimensions. 

In  measuring  length,  the  yard  derived  from  the  standard  yard  of  England 
is  the  standard  unit,  the  yards  of  the  United  States  and  England  being  iden- 
tical. Theoretically,  the  yard  is  equal  to  ff  i§ff  of  the  length  of  a  pendulum 
that  vibrates  seconds  in  a  vacuum,  at  the  level  of  the  sea,  in  the  latitude  of 
London  ;  that  is,  a  pendulum  that  vibrates  seconds  under  the  above  conditions 
is  39.1393  inches  in  length.  The  standard  yard  is,  in  fact,  the  distance  be- 
tween two  points  on  a  brass  bar,  preserved  at  Washington,  the  distance  to  be 
taken  when  the  bar  is  at  a  temperature  of  62°  Fahrenheit. 


TABLE. 


12    Inches  (in.,")=  1  Foot  . 

3    Feet  =  1  Yard  . 

5|  Yards  =  1  Rod    . 

320    Rods  =  1  Mile  . 


yd. 
rd. 
mi. 


ml.       rd.  yd.  ft. 

I  =  320  =  1760  =  5280 


1  =         3  = 
1  = 


in. 

=  63360 
=  198 
36 


12 


NOTES. — 1.  The  inch  is  usually  divided  into  halves,  quarters,  eighths,  and 
sixteenths. 

2.  The  foot  and  inch  are  divided  by  civil  engineers  and  others  into  tenths, 
hundredths,  thousandths,  etc. 

3.  In  measuring  cloth,  ribbon,  and  other  goods  sold  by  the  yard,  the  yard 
is  divided  into  halves,  quarters,  eighths,  and  sixteenths.     . 

4.  At  the  U.  S.  Custom  Houses,   the  yard  is  divided  into  tenths  and 
hundredths. 

5.  The  mile  (5280  ft.)  of  the  above  table  is  the  legal  mile  of  the  United 
States  and  England,  and  hence  it  is  sometimes  called  the  statute  mile.     . 

6.  1  furlong  =  £  mile  =  40  rods.     (Rarely  used.) 

313.  The  following  denominations  are  also  used  : 

1  Size  —  ^  Inch.     Used  by  shoemakers. 

1  Hand  =  4  Inches.     Used  in  measuring  the  height  of  horses. 

1  Fathom  —  6  Feet.     Used  in  measuring  depths  a ,  sea. 

1  Cable-length         =  120  Fathoms,  or  240  yards. 

1  Geographic  Mile  =  1.15+  Statute  Miles.     Used  in  measuring  distances 
at  sea. 

1  Knot  =  1  Geo.  Mile.     Used  in  determining  the  speed  of  vessels. 

60  Geo.  Miles,  or  \     _  J  of  latitude  on  a  meridian,  or  of  longitude 

69.16  Stat.  Miles  J    =  *  I         on  the  equator. 

360  Degrees  =  the  Circumference  of  the  Earth. 


Art.  314.] 


LINEAR     MEASURES. 


105 


314.  Surveyors'  Linear  Measure  is  used  in  measuring 
land,  roads,  etc. 

The  unit  of  this  measure  is  a  chain,  4  rods  or  66  feet  in  length,  called 
Gunter's  Chain.  It  is  divided  into  100  parts  called  links,  each  link  being  7.92 
inches  in  length. 

TABLE. 


100  Links  (1.)  =  1  Chain  . 
80  Chains      =  1  Mile 


mi.        ch.  ft.  I.  in. 

1    =   80  =  5280  =  8000  =  63360 

1  =       66  =     100  =       792 

.66  =        1  =      7.92 


NOTES. — 1.  Links  are  written  decimally  as  hundredths  of  a  chain. 

2.  For  railroad  and  other  purposes,  engineers  use  a  chain  or  tape  100  feet 
long,  the  feet  being  divided  into  tenths. 

3.  1  rod  =  25  links. 

EXAM  PLES. 

315.     1.  Add  9/tf.  8  in.,  12ft.  6  in.,  16ft.  5  in.,  loft.  11  in., 
21  ft.  4  in.,  lift.  3  in. 

2.  In  |  of  a  yard,  how  many  inches  ? 

S.  How  many  feet  in  17  miles  ?     In  35  rods  ? 

4.  Find  the  difference  between  5ft.  4^  in.  and  16J-  hands. 

5.  Reduce  49175  ft.  to  higher  denominations. 

6.  Reduce  32  rd.  4  yd.  2ft.  10  in.  to  inches. 

7.  How  many  feet  in  -J  of  a  mile  ? 

S.  In  4376  feet,  how  many  chains  ?     How  many  inches  ? 
9.  In  396  rods,  how  many  chains  ?     How  many  feet  ? 

10.  In  37.56  chains,  how  many  feet  ?     How  many  rods  ? 

11.  Children's  size  1  of  shoemakers'  measure  is  4J-  inches  long ; 
what  is  the  length  of  boys'  size  8,  youths'  size  1,  and  men's  size 
10  ?     (Size  1  of  the  second  series  is  one  size  longer  than  size  13  of 
the  first  series.) 

12.  How  many  fathoms  in  1722 ft.?     In  3136  ft.? 

13.  Reduce  48276  ft.  to  higher  denominations. 

14.  Add  4.16  ch.,  3.75  ch.,  8.08  ch.,  17.28  ch.,  46.10  ch.,  and 
38.09  ch. 

15.  How  much  will  it  cost  at  $3.25  per  rod  to  fence  a  field 
whose  sides  are  SOS  ft.,  975/tf.,  822ft.,  and"  992  ft.  respectively  ? 

16.  How  many  posts  placed  8  ft.  apart  will  be  required  to  fence 
a  railroad  14  miles  in  length  ?     How  many  feet  of  wire  will  be 
required,  the  fence  being  5  wires  high  ? 


106  DENOMINATE     NUMBERS.  [Art.  316. 


SQUARE    MEASURES. 

316.  Square   Measure  is  used  in  measuring  surfaces,  as 
land,  paving,  painting,  plastering,  roofing,  etc. 

The  unit  of  square  measure  is  a  square  bounded  by  lines  of  some  known 
length.  Thus,  a  square  inch  is  a  square  whose  sides  are  one  inch  long  ;  a 
square  foot,  a  square  whose  sides  are  one  foot  long  ;  etc. 

TABLE. 

144    Square  Inches  (sq.  in. )  =  1  Square  Foot  .     sq.  ft. 

9    Square  Feet  =  1  Square  Yard  .     .     .    sq.  yd. 

30J  Square  Yards  =  1  Square  Eod  .     .     .    sq.  rd. 

160    Square  Rods  =  1  Acre A. 

640    Acres  =  1  Square  Mile  .     .     .  sq.  mi. 

NOTES. — 1.  1  Rood  =  40  sq.  rds.  =  \  A.  The  rood  has  practically  gone 
out  of  use. 

2.  All  of  the  above,  excepting  the  acre,  are  derived  from  the  corresponding 
units  of  Linear  Measure.     Thus,  1  sq.  ft.  =  144  (12  x  12)  sq.  in. ;  1  sq.  yd.  = 
0  (3  x  3)  sq.  ft. ;  1  sq.  yd.  =  30^  (5£  x  5*)  sq.  rd. 

3.  The  acre  is  the  common  unit  of  land  measure,  and  is  equivalent  to  a 
square  whose  side  is  208.71  feet,  or  a  rectangle  10  rods  by  16  rods  (165  ft. 
by  264  ft.). 

4.  Roofers,  plasterers,  and  carpenters  sometimes  call  100  square  feet  a 
square. 

317.  Surveyors'  Square  Measure  is  used  in  measuring 

land. 

TABLE. 

10000  Square  Links  (sq.  1.)  =  1  Square  Chain  .    .    .  sq.  cli. 
10  Square  Chains  =  1  Acre A. 

NOTE. — In  the  vicinity  of  St.  Louis,  and  in  other  parts  of  the  Mississippi 
valley  that  were  settled  by  the  French,  the  old  French  arpent  is  still  used  as 
the  unit  of  land  measure.  It  contains  about  f  of  an  English  acre. 

318.  U.  S.  Public  Lands  are  divided  by  north  and  south 
lines  run  according  to  the  true  meridian,  and  by  others  crossing 
them   at   right   angles,    so   as   to   form   townships   of   six   miles 
square. 

Townships  are  subdivided  into  sections,  containing,  as 
nearly  as  may  be,  640  acres  each,  or  1  square  mile. 

Sections  are  subdivided  into  half -sections,  quarter-sections, 
half -quarter-sections,  and  quarter-quarter-sections. 


Art.  3 IS.] 


SQUARE    MEASURES, 


107 


1  Township 

1  Section 

1  Half-Section 

1  Quarter-Section 

1  Half-Quarter-Section 


TABLE. 

=  6  mi.  x  6  mi.  =  36  sq.  mi.  —  23040  A. 
=  1  "    xl  "   =  1       "       =      640 " 
=  I  "    x-|-  "  =  I      "      —      320" 
=  ±  "    xj-  "  =  J      "      =      160" 
—  J-  "    x-J  "   =  i      "      —        80" 


1  Quarter-Quarter-Section  =  J  "    xi  "  =  TV  =        40" 

The  following  diagrams  show  the  method  of  numbering  the  sections  of  a 
township,  and  that  of  naming  the  subdivisions  of  sections. 


A  TOWNSHIP. 
N 


A  SECTION. 
N 


6 

5 

8 

4 
9 
16 

3 
10 

2 

11 

1 

7 

12 
13 
24 
25 

18 

17 
20 
29 

15 

14 

19 
30 
81 

21 

22 

23 

28 

27 
34 

26 

32 

33 

35 

3d 

X.i 
320  A. 

N.W.  i 
of 
S.W.  i 

40  yl. 

EJ 
of 

S.W.  i  • 
80-4. 

S.J 

-w 

j_ 

!£ 

S.W.  1 
of 

S.W.  i 
40  A. 

319.  A  Rectangle  is  a  plane  (flat)  surface  having  four  straight 
sides  and  four  square  corners  (right  angles). 

A  rectangle  whose  sides  are  equal  is  called  a  square. 

02 O.  The  Area  of  a  surface  is  an  expression  for  that  surface 
in  terms  of  square  units. 

In  the  diagram  each  small  square 
represents  a  square  foot.  Since  there  are 
8  rows,  and  4  square  feet  in  each  row, 
there  are  3  times  4  square  feet,  or  12 
square  feet  in  the  rectangle.  Hence,  the 
area  of  any  rectangle  may  be  found  by 
multiplying  together  the  numbers  denot- 
ing its  length  and  breadth,  in  the  same 
denomination  ;  or,  more  briefly, 


4  feet. 


To  find  the  area  of  a  rectangle,  multiply  its  length  by 
Us  breadth. 


108  DENOMINATE     NUMBERS.  [Art.  391. 

EXAMPLES. 

321.     1.  Reduce  28140  square  rods  to  acres. 

2.  How  many  square  feet  in  3  acres  ? 

3.  Reduce  4  A.  100  sq.  rd.  20  sq.  yd.  to  square  yards. 

4.  Reduce  46.3125  A.  to  integers  of  lower  denominations. 

5.  How  many  acres  in  the  State  of  Wisconsin,  whose  area  is 
53924  square  miles  ? 

6.  How  many  square  feet  in  a  lot  25  feet  front  and  100  feet 
deep  ?     (32O.) 

7.  How  many  square  feet  in  a  roof  20  ft.  wide  and  45  ft.  long? 

8.  How  many  square  feet  in  a  floor  42  feet  long  and  33  feet 
wide  ?     How  many  square  yards  ? 

9.  How  many  square  feet  in  a  tight  board  fence  8ft.  high  and 
120  ft.  long  ? 

10.  How  many  building  lots,  each  36  ft.  by  110  ft.,  can  be  made 
from  a  lot  containing  5  acres  ? 

11.  How  many  acres  in  a  farm  384  rods  long  and  245  rods 
wide  ? 

12.  How  many  acres  in  a  rectangular  field   28.50  chains  by 
46.38  chains  ? 

13.  How  many  acres  in  a  rectangular  piece  of  land  224  links 
by  448  links  ? 

14.  How  many  square  yards  in  a  floor  16  ft.  6  in.  by  12  ft. 
9  in.  ? 

15.  How  many  square  feet  of  floor  in  a  3-story  building  QQft. 
by  98/1? 

16.  A  ceiling,  whose  area  is  720  sq.  ft.,  is  30  ft.  long.     What 
is  its  width  ? 

17.  What  is  the  value  of  a  field  320  rd.  long  and  160  rd.  wide, 
at  $22.50  an  acre  ? 

18.  A  rectangular  lot  contains  24  acres  ;  what  is  its  width,  its 
length  being  1056  feet  ? 

19.  What  part  of  a  square  foot  is  a  surface  3  in.  by  8  in.  ? 
4  in.  by  9  in.  ?     8  in.  by  12  in.  ? 

20.  How  many  square  yards  of  oil-cloth  will  cover  a  floor  15  ft. 
long,  13$  ft.  wide  ? 

21.  A  railroad  passes  through  5808  feet  of  a  farm.     If  the  area 
occupied  is  50  ft.  wide,  what  is  the  cost  of  the  right  of  way  at  $66 
per  acre  ? 


Art.  321.]  SQUARE    MEASURES.  109 

22.  How  many  square  feet  in  60  panes  of  glass  each  24  in.  by 
30  in.  ? 

23.  How  many  square  yards  of  paving  in  a  street  1200  ft.  long 
and  60/2.  wide? 

24.  How  many  square  feet  in  a  sidewalk  6  ft.  wide  and  |  of  a 
mile  long  ? 

25.  What   part   of  a  square  yard  is  a  surface  6  in.  x  8  in.  ? 
10  in.  x  14  in.  ?    14  in.  x  20  w.  ? 

#0.  How  many  shingles,  3  in.  wide  and  4  in.  of  the  length 
exposed  to  the  weather,  would  be  required  for  a  square  yard  of 
roofing  ? 

27.  How  many  shingles,  5  in.  wide  and  4  w.  of  the  length 
exposed  to  the  weather,  would  be  required  for  a  roof  60  ft.  long 
and  24  ft.  wide  ? 

#£.  How  many  square  feet  in  a  piece  of  tin  20  in.  x  14  in.  ? 

29.  How  many  pieces  of  tin,  14  in.  x20  in.,  will  be  required 
for  a  roof  60  ft.  long  and  49  ft.  wide,  making  no  allowance  for 
seams  and  waste  ? 

30.  How   many  brick,  upper  surface  4  tw.  x  8  in.,  will   be 
required  for  a  walk,  6/£.  wide  and  660  ft.  long  ? 

31.  How  many  paving  stones,  6  in.  by  8  m.,  will  be  required 
for  a  street  50  ft.  wide  and  1248  ft.  long  ? 

##.  Find  the  value  of  a  quarter-section  (318)  of  land  at  $6.50 
per  acre. 

33.  |  of  the  land  in  a  western  township  (318)   is  assessed  at 
$6  per  acre  and  the  remainder  at  $8.    What  is  the  total  assessment  ? 

34.  How  much  will  it  cost  to  build  a  road  from  the  common 
corner  of  sections  6,  5,  7,  and  8  (see  diagram,  Art.  318)   to  the 
common  corner  of  sections  2,  1,  12,  and  11  at  $1.25  per  linear 
rod  ?     How  many  acres  of  land  will  be  occupied  if  the  road  is  4 
rods  wide  ? 

35.  How  many  square  feet  in  the  walls  of  a  room,  16/V.  wide, 
IS  ft.  long,  and  9ft.  high  ? 

NOTE.— There  are  2  ends  each  16  ft.  x  9  ft.,  and  2  sides  each  18  ft.  by  9  ft. 
The  following  method  is  frequently  used  by  mechanics  :  Multiply  the  pe- 
rimeter (the  distance  around  the  room)  by  the  height.  To  find  the  perimeter, 
add  twice  the  length  to  twice  the  width. 

36.  How  many  square  inches  of  gold  leaf  would  be  required  to 
cover  a  box  7  in.  x  4  in.  X  3  in..?     (Use  crayon  box  as  an  illus- 
tration.) 


110  DENOMINATE     NUMBERS.  [Art.  321. 

37.  How  many  square  yards  of  plastering  surface  in  the  sides 
and  ends  of  a  room,  9  ft.  high,  16  ft.  long,  15  ft.  wide  ?     How 
many  square  yards  in  the  ceiling  ? 

38.  How  many  square  yards  of  plastering  surface  in  a  room 
12  ft.  high,  20  ft.  long,  and  18  ft.  wide,  deducting  120  sq.  ft.  for 
doors  and  windows  ? 

39.  How  many  square  feet  of  painting  surface,  excepting  the 
bottom,  011  the  outside  of  a  car,  '30ft.  long,  8ft.  wide,  7  ft.  high  ? 

40.  How  many  square  feet  of  tin  plate  would  be  required  for 
making  1000  rectangular  cans  8  in.  x  6  in.  x  15  in.,  adding  7  sq. 
in.  for  seams  and  waste  in  making  each  can  ? 

41.  How   many   yards   of  paper  border,  5   strips  in  a  piece, 
would  be  required  for  a  room  16  ft.  3  in.  long  and  12  ft.  6  in. 
wide,  adding  2ft.  6  in.  for  chimney,  jambs,  etc.? 

42.  How  many  squares  (100  square  feet)  in  a  roof  40  ft.  by 
60 ft.?    45 ft.  by  64/f.f 

NOTE. — Divide  by  100  by  pointing  off  2  figures  at  the  right. 

43  How  many  shingles,  exposed  portion  5  in.  by  5  in.,  in  a 
square  cf  100  square  feet  ?  If  exposed  portion  is  4  in.  by  4  in., 
how  many  ?  4  in.  by  5  m.,  how  many  ? 

,£4.  How  many  sheets  of  tin  20  in.  by  14  in.  in  a  square  of  100 
square  feet  ?  10  in.  by  14  in.  ?  8  in.  by  10  in.  ? 

^5.  How  many  shingles  would  be  required  for  a  roof  60  ft.  by 
80  //.,  if  500  shingles  will  cover  a  square  ?  How  many,  TOO 
shingles  to  a  square  ?  How  many,  800  shingles  to  a  square  ? 

46.  How  many  sheets  of  tin  would  be  required  for  a  roof  30ft. 
by  50ft,,  if  50  sheets  will  cover  a  square  ?     How  many,  62  sheets 
to  a  square  ? 

47.  What  part  of  a  square  yard  in  a  piece  of  carpet  27  in.  wide 
and  1  yd.  long  ? 

48-  How  much  will  it  cost  to  carpet  a  floor  15ft.  by  18ft.  with 
carpeting  j-  yd.  wide,  at  $1.60  per  yard,  making  no  allowance  for 
waste  in  matching  figures,  etc.  ? 

NOTE. — If  no  allowance  is  made  for  waste  in  cutting,  divide  the  number 
of  square  yards  in  the  floor  by  the  number  of  square  yards  in  one  linear  yard 
of  carpet.  Carpet  dealers  in  estimating  the  number  of  yards  of  carpet 
required  for  a  room,  multiply  the  length  of  the  room  (plus  a  proper  allowance 
for  matching  the  design)  by  the  number  of  full  widths.  Ingrain  carpet  is 
usually  1  yd.  wide,  Brussels,  Moquette,  Wilton,  Velvet,  and  Axminster,  f  yd. 
wide. 


Art.  321.]  SQUARE    MEASURES.  Ill 

£9.  How  many  whole  widths  of  carpet  1  yd.  wide  would  be 
required  for  a  room  14//.  8  in.  wide,  and  how  many  inches  would 
be  folded  under  ?  If  the  room  is  19  ft.  6  in.  long,  how  many 
yards  of  carpet  would  be  required,  supposing  the  excess  at  the 
sides  to  be  folded  under,  and  2£  yds.  to  be  wasted  in  cutting  and 
matching  the  figures  ? 

50.  How  many  yards  of  carpet  border  would  be  required  for  a 
room  21  ft.  by  lG%ft.? 

51.  The  height  of  a  flight  of  stairs  is  12  ft.     How  many  steps, 
if  they  are  each  8  in.  high  ?     How  many  yards  of  carpet  would  it 
be  necessary  to  purchase  if  the  tread  of  each  step  is  10  in.,  and 
allowing  one  yard  for  moving  ?     (Find  what  part  of  a  yard  is 
required  for  one  step.) 

52.  How  many  square  yards  in  a  roll  of  paper  8  yd.  long  and 
18  in.  wide  ?     How  many  sq.  ft.  ? 

53.  How  many  rolls  of  paper,  8  yd.  long,  18  in.  wide,  would 
be  required  for  the  sides  and  ends  of  room  22  ft.  6  in.  long,  13  ft. 
6  in.  wide,  and  9  ft.  high,  deducting  12  sq.  yd.  for  doors  and 
windows,  and  making  no  allowance  for  waste  in  cutting  ? 

NOTE. — If  no  allowance  is  made  for  waste  in  cutting,  divide  the  surface 
to  be  papered  by  the  number  of  square  feet  (or  square  yards)  in  one  roll  of 
paper. 

In  practice,  there  is  a  great  deal  of  waste  in  cutting  and  matching  wall 
paper.  If  the  room  is  9  ft.  high,  but  two  whole  strips  could  be  cut  from  a 
roll  8  yds.  long.  If  double  rolls  (16  yds.  each)  are  used,  5  whole  strips  could 
be  cut  from  each  roll.  It  is  therefore  more  economical  to  use  double  rolls. 

Paper-hangers  in  estimating  the  number  of  rolls  required  for  a  room,  cal- 
culate the  number  of  full  strips  that  will  be  necessary  for  the  regular  surface 
of  the  walls,  and  divide  this  number  by  the  number  of  whole  strips  that  can 
be  cut  from  one  roll.  The  ends  of  rolls  are  used  for  the  surface  above  the 
doors,  and  above  and  below  the  windows,  and  other  irregular  places. 

54.  How  many  strips  of  paper,  18  in.  wide,  would  be  required 
for  a  surface  %^ft.  wide  ? 

55.  How  many  whole  strips  of  paper,  8  ft.  9  in.  long,  could 
be  cut  from  a  roll  of   paper,   8  yd.  long  ?     How  many  from  a 
double  roll,  16  yd.  long  ? 

56.  How   many   rolls  of    paper  (16  yd.   long,   18  in.    wide) 
would  be  required  for  the  sides  and  ends  of  a  room,  20  ft.  long, 
16/7.  wide,  and  Sft.  6  in.  high,  deducting  31  ft.  for  the  width  of 
doors,  windows,  mantels,  etc.?    (Paper-hangers'  method.) 


112 


DENOMINATE     NUMBERS. 


[Art.  322. 


SOLID    OR     CUBIC     MEASURE. 

322.  Solid  or  Cubic  Measure  is  used  in  measuring  solids, 
or  bodies  which  have  length,  breadth,  and  thickness  or  depth. 

The  unit  of  cubic  measure  is  a  cube,  each  of  whose  edges  is  a  unit  of  some 
known  length.  Thus,  a  cubic  inch  is  a  cube,  each  of  whose  edges  is  one 
inch  ;  a  cubic  foot  is  a  cube,  each  of  whose  edges  is  one  foot ;  etc. 


TABLE. 

1728  Cubic  Inches  (cu.in.)  =  1  Cubic  Foot  . 
27  Cubic  Feet  =  1  Cubic  Yard  . 

128  Cubic  Feet  =  1  Cord  . 


.  cu.ft. 

.  cu.  yd. 

.  cd. 


NOTES. — 1.  The  above  units,  excepting  the  cord,  are  derived  from  the 
corresponding  units  of  linear  measure.  Thus,  1  cubic  foot  contains  1728 
(12  x  12  x  12)  cubic  inches  ;  1  cubic  yard,  27  (3  x  3  x  3)  cubic  feet. 

2.  The  U.  S.  measurement  ton  for  freight  contains  40  cubic  feet. 

3.  The  U.  S.  register  tonnage  (entire  internal  cubical  capacity)  of  vessels 
is  expressed  in  tons  of  100  cubic  feet  each. 

4.  A  perch  of  masonry  is  1  rod  long,  1|  feet  thick,  and  1  foot  high,  and 
is  equal  to  24£  (16£  x  1|  x  1)  or  about  25  cubic  feet. 

323.  The  Volume  or  Solid  Contents  of  a  solid  is  an 
expression  for  that  solid  in  terms  of  cubic  or  solid  units. 


4  Feet 


\\ 


The  diagram  represents  a  solid  4  feet  long,  3  feet  broad,  and  2  feet  thick. 
Each  small  cube  is  a  cubic  foot. 
Since  the  end  of  the  solid  contains 
(3  x  2)  6  square  feet  of  surface,  it 
is  evident,  if  a  section  1  foot  thick 
be  cut  off  from  this  end,  it  can  be 
divided  into  6  cubes,  with  edges  1 
foot  in  length,  and  therefore  the 
section  will  contain  6  cubic  feet ; 
and  since  the  whole  solid  is  4  feet 
long,  and  contains  4  like  sections, 
it  must  contain  4  times  6  cubic 
feet,  or  twenty-four  cubic  feet. 
Hence  the  volume  of  a  rectangular 
solid  may  be  found  by  multiplying 
together  the  numbers  expressing 
its  length,  breadth,  and  thickness,  in  the  same  denomination ;  or,  more 
briefly, 

To  find  the  volume  of  a  rectangular  solid,  multiply 
together  its  length,  breadth,  and  thickness. 


Art.  324.]  CUBIC     MEASURE.  113 

324:.  A  Rectangular  Solid  is  a  solid  having  six  rectangular 
sides  or  faces. 

A  Cube  is  a  rectangular  solid  whose  sides  are  six  equal  squares. 
EXAM  PLES. 

325.     1.  How  many  cords  in  15744  cubic  feet  ? 

2.  How  many  cubic  inches  in  175  cubic  feet  ? 

3.  Eeduce  37368  cubic  feet  to  cubic  yards. 

4.  Add  7  cd.  49  cu.  ft.,  13  cd.  92  cu.  ft.,  12  cd.  28  cu.  ft., 
16  cd.  110  cu.ft.,  3  cd.  16  cu.  ft.,  14  cd.  80  cu.  ft. 

5.  How  many  cubic  yards  in  an  excavation,  42  ft.  long,  40  ft. 
wide,  and  9  ft.  deep  ? 

6.  How  much  will  it  cost  to  dig  a  cellar  36  ft.  long,  30  /rf. 
wide,  and  6  ft.  deep,  at  30  cents  per  cubic  yard  ? 

7.  What  is  a  pile  of  wood,  &0//.  long,  4=  ft.  wide,  and  7/£.  6  in. 
high,  worth  at  15.75  per  cord  ? 

#.  If  a  pile  of  bark  is  40  ft.  long  and  4  /7.  wide,  how  high 
must  it  be  to  contain  10  cords  ? 

9.  How  many  cords  in  a  pile  of  bark,  22.5ft.  long,  4:  ft.  wide, 
and  4.  8  /£.  high? 

j?0.  How  many  cubic  feet  in  a  box,  4/#.  6  in.  high,  8/tf.  long, 
and  3ft.  9  w.  wide? 

11.  How  many  cubic  inches  in  a  rectangular  cistern,  6  ft.  x 


.?#.  How  many  tons  in  a  shipment  which  occupies  a  space,  16  ft. 
by  Uft.  by  28/f.  ?  (322,  2.) 

.?#.  What  is  the  freight  of  350  cu.  ft.  of  merchandise  at  $8  per 
ton  ?  At  50  shillings  per  ton  ? 

14-  How  many  cubic  feet  in  a  vessel  whose  measurement  is 
2135  tons?  (322,  3.) 

15.  How  many  perch  of  masonry  in  a  wall,  40  ft.  long,  9  ft. 
high,  and  Uft.  thick  ?  (1  Perch  =  25  cu.  ft.)  (322,4.) 

.#>.  How  many  bricks  2  -iw.  x  4  m.  x  8  in.  in  one  cubic  foot  ? 

77.  The  space  occupied  by  a  bag  containing  1000  standard 
silver  dollars  is  12  in.  long,  9  in.  wide,  and  4  tw.  deep.  How 
many  cubic  feet  would  be  occupied  by  1,000,000  such  dollars  ? 

18.  How  many  cubic  feet  is  a  wall,  12  in.  thick,  42  ft.  long, 
and  30ft.  high  ?  How  many  bricks  would  be  required  for  the 
above  wall  allowing  21  to  a  cubic  foot  ?  What  would  be  their 
value  at  $9  per  thousand  ? 


114  DENOMINATE     NUMBERS.  [Art.  325. 

Common  North  River  brick  are  8  in.  x  4  in.  x  2|  in.  Brick  manufactured 
in  other  localities  are  of  various  sizes. 

Builders  usually  allow  7  common  bricks  for  each  square  foot  of  the  surface 
of  the  wall  if  the  wall  is  one  brick  thick  (4  in.),  14  to  a  square  foot  if  2  bricks 
thick  (about  8  in.),  21  to  a  square  foot  if  3  bricks  thick  (about  12  in.),  etc. 

19.  According  to  the  above  builder's  rule,  how  many  bricks 
would  be  required  for  a  wall,  64  ft.  long,  39  ft.  high,  and  2  bricks 
thick  (about  8  in.)? 

20.  How  many  cubic  feet  of  masonry  in  a  cellar  wall  2  ft. 
thick,  8  ft.  high,  outside  measurement  25  ft.  by  45  ft  ? 

Builders  sometimes  multiply  the  total  outside  measurement  (perimeter  or 
girth)  by  the  height  and  thickness  to  find  the  number  of  cubic  feet.  By  this 
method,  the  corners  are  counted  twice.  To  find  the  exact  length  of  the  wall, 
from  the  total  outside  measurement  subtract  four  times  the  thickness  of  the 
wall.  (See  note,  Ex.  35,  Art.  321.) 

21.  How  many  bricks  will  be  required  to  build  a  chimney  28ft. 
high,  if  there  are  5  courses  of  brick  to  each  foot,  and  1C  bricks  in 
each  course  (flue  4  in.  by  12  in.,  double  wall)  ?     How  many,  8 
bricks  in  a  course  (flue  8  in.  by  16  in.,  single  wall)  ? 

22.  How  much  stone,  lime,  and  sand  will  be  required  for  a 
wall  144  ft.  by  10  ft.  by  2  ft.,  if  128  cu.  ft.  of  broken  stone,  1J 
bbls.  of  lime,  and  a  load  (cubic  yard)  of  sand  will  lay  100  cu.  ft. 
of  wall  ? 

23.  How  many  bricks  will  be  required  for  a  house  24  ft.  wide, 
63  ft.  long,  and  30  ft.  high,  allowing  21  bricks  for  each  square 
foot  of  surface  (the  walls  being  3  bricks  thick),  if  225  square  feet 
are  deducted  for  doors  and  windows,  and  if  the  walls  are  con- 
sidered 1  ft.  thick  in  making  deductions  for  the  corners  ? 

BOARD     MEASURE. 

326.  Lumber  is  measured  by  board  measure.  The  board 
foot  is  1  ft.  long,  1  ft.  wide,  and  1  in.  thick ;  hence  it  is  -^  of  a 
cubic  foot. 

In  measuring  boards  one  inch  or  less  in  thickness,  the  number  of  square 
feet  of  surface  which  they  would  cover  is  measured. 

Plank,  joists,  etc.,  more  than  one  inch  in  thickness  are  reduced  to  inch 
boards  and  measured  by  board  measure. 

Boards,  plank,  scantling,  joists,  beams,  and  sawed  timber  generally  are 
measured  by  board  measure  ;  hewn  and  round  timber  are  sometimes  measured 
by  cubic  measure. 


Art.  327.]  BOARD     MEASURE.  115 

327.  When  lumber  is  not  more  than  one  inch  thick,  to  find 
the  number  of  feet  board  measure  :   Multiply  the  length  in  feet  by 
the  width  in  inches,  and  divide  the  product  by  12. 

16  pieces  each  containing  a  board  foot  could  be  cut  from  a  board  16  ft. 
long,  12  in.  wide.  A  board  11  in.  wide  is  {%  of  a  board,  12  in.  wide  and  of 
the  same  length  and  thickness. 

When  more  than  1  inch  thick  :  Multiply  the  length  in  feet  by 
the  width  and  thickness  in  inches,  and  divide  the  product  by  12. 

Lumbermen  use  an  automatic  rule  for  measuring  board,  planks,  etc.  They 
also  use  table  books  for  reducing  logs  or  round  timber  to  board  measure. 

EX  A  M  PLES. 

328.  1.  How  many  square  feet  would  be  covered  by  a  board 
16  ft.  long  and  9  in.  wide  ? 

2.  If  a  stick  of  timber  24  ft.  long,  8  in.  wide,  3  in.  thick,  is 
reduced  to  boards  one  inch  in  thickness,  how  many  square  feet 
would  they  cover  ? 

3.  How  many  feet  of  boards  would  be  required  for  a  floor 
20ft.  wide  and  24  ft.  long  ?     What  would  be  their  value  at  $14 
per  thousand  ? 

4.  How  many  board  feet  in  475  cubic  feet  ? 

Find  the  number  of  feet,  board  measure,  in  each  of  the  follow- 
ing boards,  joists,  beams,  etc. : 


Length. 

Width.     Thickness. 

Length. 

Width. 

Thicknees. 

5. 

12 

A 

12 

in. 

1 

in. 

9. 

16  ft. 

8 

in. 

1J  in. 

6. 

14 

A 

8 

in. 

i 

in. 

10. 

20  ft. 

10 

in. 

2  in. 

7. 

16 

A 

6 

in. 

1 

in. 

11. 

12  ft. 

6 

in. 

4  in. 

8. 

18 

A 

14 

in. 

i 

in. 

12. 

24  ft. 

9 

in. 

3  in. 

13.  How  many  square  feet  would  be  covered  by  45  boards, 
16  ft.  long,  8  in.  wide,  if  they  are  laid  side  by  side  ?     (First  find 
the  total  width  in  feet.) 

14.  How  many  board  feet  in  a  pile  of  lumber  8  ft.  high,  7  ft. 
wide,  and  18  ft.  long  ?    Find  its  value  at  $15  per  thousand  feet. 

15.  How  many  feet,  board  measure,  in  16  boards,  each  18  ft. 
long,  10  in.  wide,  and  1  in.  thick  ? 

16.  How  many  board  feet  in  24  joists,  20  ft.  x  8  in.  x  4  in.  ? 

17.  How  many  feet  of  boards,   4  in.   wide  and  1  in.  thick, 
would  be  required  for  a  fence  5  boards  high  and  one  mile  long  ? 

18.  How  many  posts  8ft.  apart  would  be  required  for  above 
fence  ? 


116 


DENO  MINA TE     N UMB E R S . 


[Art.  328. 


19.  How  many  feet,  board  measure,,  in  13  planks,  each  10  ft. 
long,  12  in.  wide,  and  2  in.  thick  ? 

*    20.  Making  no  allowance  for  the  corners,  how  many  feet  of 
boards  would  be  required  to  make  a  box  8ft.  x  4=  ft.  x  oft.  ? 

21.  How  many  feet  in  a  tapering  board  18/2.  long,  12  in.  wide 
at  the  smaller  end,  and  16  in.  at  the  other  ? 

NOTE. — To  find  the  average  width  of  a  tapering  board,  measure  it  at  the 
center,  or  take  \  the  sum  of  the  widths  at  the  ends. 


gal. 

qt. 

pt. 

gi. 

.  pt. 

-j    

4 

Q       

32 

.      qt. 

1 

=   2  = 

8 

.  gal. 

1   = 

4 

LIQUID     MEASURES. 

329.  Liquid  Measure  is  used  for  measuring  liquids. 

The  unit  of  this  measure  is  the  wine  gallon,  which  contains  231  cubic 
inches. 

TABLE. 

4  Gills  (gi.)  =  I  Pint     . 
2  Pints  =  1  Quart  . 

4  Quarts        =  1  Gallon 

NOTES. — 1.  In  estimating  the  capacity  of  tanks,  cisterns,  reservoirs,  etc., 
1  barrel  =  31^  gallons  ;  1  hogshead  =  2  barrels  =  63  gallons. 

2.  In  commerce,  the  barrel,  tierce,  and  hogshead  are  not  fixed  measures, 
but  their  capacity  is  found  by  gauging,  or  actual  measurement. 

3.  The  imperial  gallon  of  England  contains  277.274  cubic  inches,  and  is 
equivalent  to  1.2  U.  S.  wine  gallons. 

4.  1   cubic   foot  =  7.48  or  about   7£  (1728  -5-  231  =  7.48)  wine  gallons. 
Hence  to  find  the  number  of  gallons  in  a  rectangular  cistern,  multiply  the 
number  of  cubic  feet  by  7.48  or  7^. 

5.  To  find  the  number  of  gallons  in  a  cylindrical  vessel,  multiply  the  square 
of  the  diameter  by  the  height,  and  this  product  by  5|  (.7854x1728-^-231  = 
5.8752). 

330.  Apothecaries'  Fluid  Measure  is  used  in  prescribing 
and  compounding  liquid  medicines. 

The  gallon  and  pint  of  this  measure  are  the  wine  gallon  and  pint. 


TABLE. 

60  Minims  (Tit)    =  1  Fluidrachm     .    /  3  . 

8  Fluidrachms  =  1  Fluidounce     .    /  §  . 
16  Fluidounces  =  1  Pint       ...        0. 

8  Pints  =  1  Gallon    .     .     .  Cong. 


Cvng.  0.  f  I .   f  3  .    % 

1  =  8  =  128  =  1024  =  61440 

1  =  16  =  128  =  7680 

1  =   8  =   480 

1  =   GO 


Art.  330.]  LIQUID     MEASURES.  117 

NOTES. — 1.  Cong,  is  for  the  Latin  congius,  gallon  ;  0.,  for  the  Latin  octar 
vius,  one-eighth. 

2.  The  symbols  precede  the  numbers  to  which  they  refer  ;  thus,  0.  6 
/5  10,  is  6  pints  10  fluidounces. 

EXAMPLES. 

331.     1.  Reduce  8  gal  3  qt.  1  pt.  to  pints. 

2.  Reduce  875  pints  to  gallons. 

3.  Add  4  gal.  2  qt.  Ipt.,  3  gal.  3  qt.  I  pt.,  9  gal.  1  pt.,  11  gal. 
1  qt. 

4.  How  many  barrels  in  100000  gallons  ? 

5.  480  English  gallons  (329,  3)  equal  how  many  U.  S.  gal- 
lons ?     (Add|.) 

6.  How  many  cubic  feet  in  1000  gallons  ? 

7.  How  many  gallons  in  300300  cubic  inches  ? 

8.  How  many  gallons  in  a  rectangular  tank,  8ft.  x  4//.  x  6ft.  ? 

OPERATION. — 8x4x6x7^  =  1440,  approximate  result  (see  Art.  329, 
Note  4).  8x4x6x7. 48  =  1436.16,  accurate  result. 

9.  How  many  gallons  in  a  rectangular  cistern  6  ft.  long,  4  ft. 
wide,  3ft.  high? 

10.  How  many  gallons  in  a  rectangular  cistern  16  ft.  long,  4 
ft.  high,  and  6ft.  wide  ? 

11.  How  many  gallons  in  a  rectangular  reservoir,  40  ft.  long, 
16ft.  wide,  and  8ft.  deep  ? 

./£.  How  many  gallons  in  a  cylindrical  vessel,  3ft.  in  diameter 
and  9 ft.  high? 

OPERATION.— 3  x  3  x  9  x  5|  =  475f  (see  Art.  329,  Note  4). 

13.  How  many  gallons  in  a  cylindrical  tank  4:  ft.  in  diameter 
and  16ft.  deep? 

!£•  How  many  gallons  in  a  circular  reservoir,  40  ft.  in 
diameter  and  6ft.  deep  ? 

.75.  How  many  gallons  in  a  circular  reservoir  60  ft.  in  diameter 
and  8ft.  deep  ? 

^<5.   Reduce  Cong. 2  0.6  / 3  10  /3  5  to  fluidrachms. 

17.  From  the  sum  of  51  ^aJ.  2  qt.  1  pt.,  and  45  gal.  1  gtf.  1  pt.9 
subtract  27  ##?.  1  qt.,  and  divide  the  result  by  9. 

18.  How  many  bottles,  each  holding  1  qt.  1  pt.  2  gi.,  can  be 
filled  from  a  barrel  of  cider  ? 


118  DENOMINATE     NUMBERS.  [Art.  332. 


DRY     MEASURE. 

332.  Dry  Measure  is  used  in  measuring  dry  articles;  as 
salt,  grain,  fruits,  etc. 

The  unit  of  this  measure  is  the  Winchester  bushel,  which  contains  2150.42 
cubic  inches. 

TABLE. 


2  Pints  (pt.}  =  1  Quart  .  .  .  qt. 
8  Quarts  =  1  Peck  .  .  .  pk. 
4  Pecks  =  1  Bushel .  .  bu. 


bu.        pk.          qt.          pt. 

1   m  4  =   32  =   64 

1   =     8  =   16 


NOTES. — 1.  The  half-peck  or  gallon  of.  this  measure  contains  268.8  cubic 
inches,  and  is  37.8  cubic  inches  larger  than  the  liquid  gallon  (268.8  — 
231  =  37.8). 

2.  The  imperial  bushel  of  England  contains  2218.19  cubic  inches,  and  is 
equal  to  1.03  Winchester  bushels.     In  certain  localities  in  Great  Britain,  8 
bushels  are  called  a  quarter. 

3.  Grain,  seeds,  etc.,  are  usually  sold  by  weight.     Portable  of  equivalents 
see  Art.  338. 

4.  36  bushels  =  1  chaldron  of  coke  or  charcoal. 

5.  1  bushel  is  equivalent  to  about  9.3  (2150.42  -4-  231)  wine  gallons. 

EXAMPLES. 

333.     1.  Reduce  2  bu.  3  pk.  5  qt.  1  pt.  to  pints. 

2.  Reduce  10000  pints  to  bushels. 

3.  Find  the  value  of  5  bushels  of  nuts  at  Sc.  per  pint. 

4.  How  many  cubic  inches  in  75  bushels  ? 

5.  How  many  bushels  in  322563  cubic  inches  ? 

6.  How  many  bushels  in  400  cubic  feet  ? 

NOTE. — Since  a  bushel  is  about  1£  cubic  feet,  the  following  approximate 
rules  may  be  used  for  all  practical  purposes  : 

To  reduce  cubic  feet  to  bushels:   Deduct  one-fifth,  or  multiply  by  .8 

The  result  will  be  too  small  by  about  4£  bushels  for  every  1000  bushels  of 
the  result. 

To  reduce  bushels  to  cubic  feet:  Add  one- fourth,  or  divide  by  .8. 

The  result  will  be  too  great  by  about  4£  cubic  feet  for  every  1000  cubic 
feet  of  the  result. 

Solve  the  above  example  both  exactly  and  approximately,  and  compare 
the  results. 

7.  How  many  bushels  will  a  box  10/tf.  long,  5  ft.  wide,  and 
±  ft.  high  contain  ?     (Approximate  method.) 


t    ^r  /\      L^     I—- 


Art. 333.]  MEASURES     OF    WEIGHT.  119 

8.  How  many  bushels  of  grain  will  a  bin  14  ft.  long,  3J  ft. 
wide,  and  6  ft.  high  contain  ? 

9.  Find  the  capacity  in  bushels  of  a  crib,  20  ft.  long,  8  ft. 
high,  4/V.  wide  at  the  bottom  and  6  ft.  wide  at  the  top. 

NOTE. — To  find  the  average  width,  take  one-half  the  sum  of  the  top  arid 
bottom  measurements. 

10.  A  crib  24  ft.  xSft.  xQft.  is  filled  with  unshelled  corn. 
How  many  bushels  of  shelled  corn  would  this  quantity  produce, 
if  two  cubic  feet  of  corn  in  the  ear  will  make  one  bushel  of 
shelled  corn  ? 

MEASURES     OF    WEIGHT. 

334.  Troy  "Weight  is  used  in  weighing  gold,  silver,  coins* 
and  jewels  ;  also  in  philosophical  experiments. 

The  unit  of  weight  is  the  Troy  pound,  which  contains  5760  grains.  A 
cubic  inch  of  distilled  water  weighs  252.458  of  these  grains,  when  the  height 
of  the  barometer  is  30  inches,  and  the  temperature  of  the  air  and  water  62° 
Fahrenheit. 

TABLE. 


24  Grains  (gr.)  =1  Pennyweight  pwt. 
20  Pennyweights  =  1  Ounce  .  .  .  oz. 
12  Ounces  =  1  Pound  .  .  Ib. 


lb.       oz.       pwt.          gr. 
1  =  12  =  240  =  5760 

1  =    20  =    480 


NOTE. — The  carat,  used  in  weighing  diamonds,  equals  3.2  Troy  grains. 
The  term  carat  is  also  used  to  denote  the  fineness  of  gold,  and  means  ^ 
part.     Thus,  gold  18  carats  fine  contains  18  parts  pure  gold  and  6  parts  alloy. 

335.   Apothecaries'   "Weight  is  used  in  prescribing  and 
compounding  medicines  not  liquid. 

The  pound,  ounce,  and  grain  of  this  weight  are  the  same  as  those  of  Troy 
weight,  the  division  of  the  ounce  being  different. 


TABLE. 

20  Grains  (gr.)  =  1  Scruple  .  .  sc.  or  3 . 
3  Scruples  =  1  Dram  .  .  dr.  or  3  . 
8  Drams  =  1  Ounce  .  oz.  or  5  . 


ft       I         3          3         gr. 

1  =  12  =  96  =  288  =  5760 
1  =    8  =    24  =    480 
1  =      3  =      GO 


12  Ounces         =  1  Pound     .     .     lb.  or  lb . 

NOTES. — 1.  The  symbols  precede  the  numbers  to  which  they  refer  ;  thus. 
36  3  4,  is  6  ounces  4  drams. 

2.  Drugs  and  medicines  are  sold  in  large  quantities  by  Avoirdupois  weight 


120 


DENOMINATE     NUMBERS. 


[Art.  336. 


336.  Avoirdupois  "Weight  is  used  in  weighing  all  articles, 
excepting  gold,  silver,  precious  stones,  and  medicines  in  small 
quantities. 

The  Avoirdupois  pound  contains  7000  Troy  grains. 

TABLE. 


16  Ounces  (oz.} 
100  Pounds 


=     1  Pound      .     .     .     .     Ib. 
_  (  \  Hundred-weight,  or  cwt. 
~  \  1  Cental      ....     0. 
20  Hundred-weight  =     1  Ton T. 


T.    cwt.       Ib.  oz. 

1  =  20  =  2000  =  32000 
1  =    100  =    1600 
1  =       16 

NOTES. — 1.  The  ounce  is  divided  into  halves  and  quarters. 

2.  The  dram,  y1^  of  an  ounce,  is  now  little  used,  except  by  silk  manu- 
facturers. 

3.  The  Long  or  Gross  ton,  formerly  used,  contained  2240  pounds  ;  the 
hundred-weight,  112  pounds  ;  and  the  quarter,  28  pounds. 

These  weights  are  still  used  in  Great  Britain,  at  the  U.  S.  Custom  Houses, 
in  ocean  freights,  and  by  wholesale  dealers  in  coal  and  iron. 

337.  Comparison  of  Troy  and  Avoirdupois  weights. 


5760  grains    = 
7000  grains    = 


1  Ib.  Troy. 

1  Ib.  Avoirdupois. 


480  grains      =    1  oz.  Troy. 

437^  grains    =    1  oz.  Avoirdupois. 


338.  In  buying  and  selling  grain,  seeds,  and  other  produce, 
the  bushel  is  regarded  as  a  certain  number  of  pounds.  The 
Boards  of  Trade  of  the  principal  cities  of  the  United  States  use 
the  equivalents  given  in  the  following  table  : 

TABLE  OF  AVOIRDUPOIS  POUNDS  ix  A  BUSHEL. 


Commodities. 

Lbs. 

Commodities. 

Lbs. 

Commodities. 

Lbs. 

48 

Corn   shelled 

56 

Peas 

60 

Beans  

60 

Corn  in  the  ear. 

70 

Rve.  . 

56 

Buckwheat  

48 

Malt  

34 

Timothy  Seed  .  . 

45 

Clover  Seed.  .  .  , 

60 

Oats  

32 

Wheat  

60 

.  In  the  Liverpool,  San  Francisco,  and  some  other  markets,  produce  is 
bought  and  sold  by  the  cental  of  100  pounds.  Railway  freight  tariffs  in  the 
United  States  on  grain,  provisions,  etc.,  are  reckoned  per  cwt.  or  cental. 

339.  The  following  units  are  used  in  commerce  : 
1  Quintal  of  Fish  =     100  Ibs. 

I  Barrel  of  Flour  =     196  Ibs. 

1  Barrel  of  Pork  or  Beef     =     200  Ibs. 
1  Gallon  Petroleum  =       6J  Ibs. 

1  Keg  of  Nails  =     100  Ibs. 


Art.  340.]  MEASURES     OF     WEIGHT.  121 


EXAM  PLES. 

34O.     1.  Reduce  10000  grains  to  Troy  pounds. 

2.  Reduce  2  Ib.  8  oz.  16  pwt.  to  grains. 

3.  What  is  the  weight  in  Troy  ounces  of  1000  silver  dollars 

Of  1280  silver  dollars  ? 

4.  Find  the  weight  in  Troy  ounces  of  1000  gold  dollars 

5.  Find  the  weight  in  Troy  ounces  of  2000  half-dollars 

6.  What  is  the  cost  of  a  14  Jc.  watch  chain  weighing  37|  pwt. 
at  $1.15  per  pennyweight  ? 

7.  A  watch  case,   14  carats  fine,  and  weighing  60  pwt.,  con- 
tains how  many  ounces  of  pure  gold  ? 

8.  Find  the  value  of  a  diamond  weighing  ^-J-  of  a  carat,  at 
$100  per  carat. 

9.  How  many  Troy  ounces  of  pure  silver  would  be  required 
for  the  coinage  of  2,000,000  standard  silver  dollars  (113)?     How 
much  copper  ? 

10.  Reduce  fel    39    36  32  to  grains. 

11.  How  many  powders,  each  containing  5  grains,  can  be  made 
from  1  Ib.  Apothecaries  (Troy)  of   quinine  ?     How  many  from 
1  Ib.  Avoirdupois  ? 

12.  Add  8  Ib.  9  oz.,  10  Ib.  7  oz.,  14  Ib.  15  oz.,  and  17  Ib.  13  oz. 
Avoirdupois. 

18.  How  many  grains  in  16  Ib.  Avoirdupois  ? 

H.  In  70  Ib.  Avoirdupois,  how  many  pounds  Troy  ?     (337) 

15.  In  175  oz.  Troy,  how  many  ounces  Avoirdupois  ? 

16.  Find  the  cost  of  58  Ib.  4  oz.  of  butter  at  28c.  per  pound. 

17.  Find  the  cost  of  875  pounds  of  feed  at  $1.15  per  cwt.  (273) 

18.  Find  the  cost  of  17387  pounds  of  oats  at  $1.85  per  cental. 

19.  Find  the  cost  of  21370  pounds  of  hay  at  $8  per  ton. 

OPERATION. 

2  )  21370  ANALYSIS.— 21370  Ib.  =  70.685  (21370  -*-  2000)  tons.   If 

10. 685         1  ton  costs  $8»  10.685  tons  will  cost  10.685  times  $8,  or 

g         $85.48.     To  divide  by  2000,  divide  by  2  and  place  the 

point  in  the  quotient  three  places  to  the  left. 
$85.480 

20.  What  is  the  value  of  28140  pounds  of  straw  at  $6.50  per 
ton? 

21.  Find  the  cost  of  16480  pounds  of  hay  at  $11.50  per  ton. 


122  DENOMINATE     NUMBERS.  [Art.  340. 

22.  Find  the  value  of  28160  pounds  of  coal  at  $5.25  per  ton. 

23.  Find  the  value  of  42250  pounds  of  coal  at  $3. 75  per  ton  of 
2240  pounds.     (336,  3) 

How  many  bushels  in 

24.  8375  pounds  of  wheat  ?  29.  18174  pounds  of  rye  ? 

25.  9116  pounds  of  corn  ?  30.  13275  pounds  of  wheat  ? 

26.  1128  pounds  of  beans  ?  SI.  20000  pounds  of  barley  ? 

27.  5172  pounds  of  peas  ?  32.  11419  pounds  of  clover  seed  ? 

28.  3375  pounds  of  oats  ?  33.  12562  pounds  of  timothy  seed? 

84.  What  is  the  value  of  49375  pounds  of  corn  at  64c.  per 
bushel  ? 

NOTE. — Usually  in  business  computations,  the  number  of  bushels  is  also 
required.  49375  Ib.  =  881||  bu.  881  ff  times  64c.  =  $564.29. 

When  the  number  of  bushels  is  not  required,  to  avoid  fractions,  multiply 
the  number  of  pounds  by  the  price  per  bushel  and  divide  the  product  by  the 
number  of  pounds  in  one  bushel.  49375  x  $.64  -*-  56  =  $564.29. 

By  both  of  the  above  methods,  find  the  cost  of 

35.  8375  Ib.  wheat  at  $1.10  per  bushel. 

36.  9416  Ib.  corn  at  85c.  per  bushel. 

37.  7428  Ib.  oats  at  72c.  per  bushel. 

38.  6224  Ib.  beans  at  $2.25  per  bushel. 

39.  9118  Ib.  barley  at  $1.25  per  bushel. 

40.  8128  Ib.  rye  at  82c.  per  bushel. 

41.  4170  Ib.  clover  seed  at  $4.25  per  bushel. 

42.  5160  Ib.  timothy  seed  at  $1.75  per  bushel. 

43.  What  is  the  freight  on  528^-°-  bushels  corn  at  32c.  per  cwt.  ? 
44-  What  is  the  freight  of  16  T.  17  cwt.  20  Ib.  at  $5  per  ton  of 

2240  Ib.  ? 

ENGLISH     MONEY. 

341.  English  or  Sterling  Money  is  the  legal  currency  of 
Great  Britain. 

TABLE.  value  in 

U.  S.  Money. 

4  Farthings  =      1  Penny     .    .    .    d.    ...    $  .02  + 
12  Pence  =     1  Shilling  .    .    .     * 243  + 


Art.  341.]  ENGLISH    MONEY.  123 

NOTES.— 1.  1  Crown  =  5  shillings,  or  £  of  a  pound  ($1.216  +  ). 

2.  1  Guinea  =  21  shillings  ($5.11).    It  is  not  now  coined. 

3.  The  gold  coins  of  Great  Britain  are  22  carats  ({$),  or  .916f  fine.     (The 
old   carat  system  (334,  note)  is  generally  abandoned  except    for  jewelry,, 
1  carat  =  ,041f .)    The  silver  coins  of  Great  Britain  are  .925  (f£)  fine. 

EXAMPLES. 

342.  1.  Add  £27  16s.  10&,  £6  10s.  Sd.,  £47  15s.  lid.,  £25 
7s.  6d.,  £3  14s.  Sd.,  and  £23  16s.  3d. 

2.  From  £17  8s.  4=d.  subtract  £10  12s.  Sd. 

3.  Multiply  £5  6s.  3d.  by  8. 

4.  Eeduce  8375^.  to  shillings  and  pounds. 

5.  Reduce  £12  16s.  Sd.  to  pence. 

6.  What  is  the  cost  of  466  yards  of  cloth  at  9J^.  per  yard  ? 

7.  Find  the  value  of  4120  bu.  wheat  at  4s.  ^\d.  per  bushel. 

8.  In  47  guineas,  how  many  pounds  and  shillings  ? 

9.  Divide  £16  5s.  6d.  by  9  ;  by  7  ;  by  31. 

10.  How  many  yards  of  cloth  at  3s.  Id.  per  yard  can  be  bought 
for  £7  ?     For  £9  5s.  Qd.  ? 

11.  What  is  the  cost  of  20  yd.  silk  at  10s.  Qd.  per  yard  ? 

12.  Reduce  £8  17s.  Sd.  to  the  decimal  of  a  pound.     (293) 

NOTE. — The  following  method  for  reducing  shillings  and  pence  to  the 
decimal  of  a  pound  is  sufficiently  accurate  for  most  business  purposes  : 
Write  one-half  of  the  greatest  even  number  of  shillings  as  tenths,  and  if  there 
be  an  odd  shilling  write  5  hundredths  ;  multiply  the  number  of  pence  by  ^,  and 
write  the  product  as  thousandths.  If  the  product  is  between  12  and  36,  add  1 
to  the  thousandths ;  if  between  36  and  48,  add  2  to  the  thousandths.  Thus, 
£8  17s.  Sd.  =  £8  +  £.85  +  £.033  =  £8.883. 

Reduce  mentally  the  following  to  the  decimal  of  a  pound  : 

13.  16s.  Zd.  15.     10s.  Sd.  17.     7s.  3d. 

14.  18s.  5d.  16.     17s.  4rf.  18.     13s.  lid. 

19.     Reduce  £.821  of  a  pound  to  shillings  and  pence.     (289) 

NOTE. — This  example  can  be  performed  mentally  by  reversing  the  opera- 
tion explained  in  note  to  Ex.  12.  Multiply  the  number  of  tenths  by  2,  and 
write  the  product  as  shillings  (2x8  =  16).  Divide  the  number  of  thousandths 
expressed  by  the  2nd  and  3rd  figures  by  4,  and  write  the  quotient  as  pence 
(21-*- 4  =  5).  £.821  =  16s.  5d. 

If  the  second  figure  to  the  right  of  the  point  is  5  or  more  than  5,  it  is 
evident  that  there  is  an  odd  number  of  shillings,  and  the  decimal  must  be 
separated  into  two  parts  before  applying  the  above  rule. 

Thus,  £.875  =  £.85  +  £.025.     £.85  =  17s.  (2  x  8£).     £.025  =  Qd.  (25  -s-  4). 


124  DENOMINATE     NUMBERS.  [Art.  342. 

Eeduce  mentally  the  following  to  shillings  and  pence : 

20.  £.425;   £.637.  22.     £.255;   £.183. 

21.  £.817;   £.245.  23.     £.376;   £.496.       . 

'"^.Divide  £15  16*.  Sd.  by  .10;  by  .20  ;  by  .25  ;  by  .40. 

25.  Multiply  £16  12s.  9d.  by  .05  ;  by  .06  ;  .04. 

26.  If  £1  sterling  is  worth  $4.87,  what  is  the  value  of  £225 
18s.  6d.  ?     Of  £140  8s.  8d.  ? 

^Jty.  If  £1  sterling  is  worth  $4.88,  how  many  pounds  can  be 
bought  for  $1000  ?     How  many  for  $1625  ? 

MISCELLANEOUS     TABLES. 
343.  The  following  table  is  used  in  counting  certain  articles: 


12  Units  =  1  Dozen  .  .  .  doz. 
12  Dozen  =  1  Gross  .  .  .  gr. 
12  Gross  =  1  Great  Gross  .  g.yr. 


g.  gr.      gr.         doz.        units. 

I  =  12  =  144  =  1728 

1  =     12  =     144 


344.  The  following  table  is  used  in  the  paper  trade  : 


24  Sheets    =    1  Quire     ....      qr. 
20  Quires    =    1  Ream     ....     rm. 


rm.  qr.         sheets. 

1     =    20     =    480 


Manufacturers  and  wholesale  dealers  usually  sell  paper  by  the  pound. 
EXAMPLES. 

345.     1.  Find  the  value  of  5  gross  pencils  at  4c.  each. 

2.  A  merchant  buys  6  gross  pens  at  95c.  per  gross,  and  sells 
them  at  Ic.  each.     What  is  his  profit  ? 

3.  How  many  sheets  of  paper  in  12  quires  ?     In  2  reams  ? 

4*  Find  the  difference  between  six  dozen  dozen  and  half  a 
dozen  dozen. 

5.  Combs  are  bought  at  $2. 70  per  dozen.     How  much  is  that 
apiece  ? 

6.  Find  the  value  of  306  eggs  at  22c.  per  dozen. 

7.  At  1  cent  each,  what  is  the  value  of  20  great  gross  pens  ? 

8.  A  grocer  buys  81  dozen  eggs* at  22c.   per  dozen,  and  sells 
them  at  the  rate  of  9  for  25  cents.     What  is  his  profit  ? 

9.  If  32  pages  of  a  book  are  printed  on  one  sheet,  how  many 
reams  of  paper  would  be  required  for  2000  copies  containing  384 
pages  each  ? 


Art.  346.]  CIRCULAR     MEASURE.  125 


CIRCULAR     MEASURE. 

346.  Circular  or  Angular  Measure  is  used  in  measuring 
angles  and  arcs  of  circles.    It  is  employed  principally  by  surveyors 
in  determining  directions,  by  navigators  in  determining  latitude 
and  longitude  of  places,  and  by  astronomers  in  making  observa- 
tions. 

The  unit  of  this  measure  is  the  degree,  which  is  ^  of  the  circumference 
of  any  circle. 

TABLE. 

60  Seconds  (")  =  1  Minute '. 

60  Minutes          =   1  Degree °. 

360  Degrees         =  1  Circle C. 

NOTES. — 1.  A  quadrant  is  one-fourth  of  a  circle,  or  90°. 

2.  A  sextant  is  one-sixth  of  a  circle,  or  60°. 

3.  1  minute  of  the  circumference  of  the  earth  is  called  a  nautical,  or 
geographic  mile,  and  is  about  1.15  statute  or  common  miles. 

4.  An  arc  of  1  degree  of  the  circumference  of  the  earth  measured  at  the 
equator  equals  69.16  statute  miles. 

EXAMPLES. 

347.  1.  Add  74°  0'  3"  and  77°  49'  58". 
&  Add  12°  27'  14"  and  122°  26'  45". 

8.  From  84°  29'  31"  subtract  77°  0'  45". 

4.  Multiply  13°  11'  16"  by  5 ;  by  15. 

5.  Divide  76°  11;  45"  by  15  ;  by  12. 

6.  Divide  179°  42'  15"  by  15  ;  by  16. 

7.  Eeduce  1,000,000"  to  higher  denominations. 

8.  Reduce  44°  16'  40"  to  seconds. 

9.  The  angles  of  a  triangle  are  67°  18'  40",  72°  39'  50",  and 
40°  1'  30"  respectively.     What  is  their  sum  ? 


LONGITUDE    AND    TIME. 

348.  The  whole  circle  of  the  earth,  or  360°,  passes  under  the 
sun  in  24  hours,  and  in  1  hour  passes  -fa  of  360°,  or  15°;  in  1 
minute,  ^  of  15°  (15  x  60'),  or  15';  and  in  1  second,  -gV  of  15' 
(15  x  60"),  or  15". 


126 


DENOMINATE     NUMBERS. 


[Art.  349. 


349.  Comparison  of  Longitude  and  Time. 

For  a  difference  of  There  is  a  difference  of 

15°  in  Longitude  1  lir.    in  Time. 

15'    "          "  1  min.  "      " 


15"  " 
1°    " 

1'    " 


1  sec.    " 
4  min.  " 

4  sec.    " 
" 


35O.  RULE. — 1.  Tlic  difference  in  longitude  of  two  places, 
expressed  in  °  ",  divided  by  15  ivill  produce  their  differ- 
ence in  time  expressed  in  hours,  minutes,  and  seconds. 

2.  The  difference  in  solar  time  of  two  places,  expressed 
in  hr.  min.  sec.,  multiplied  by  15  will  produce  their  differ- 
ence in  longitude  expressed  in  °  '  ". 

351.  TABLE  OF  LONGITUDES. 


Albany  

73°  44'  50"  W 

New  York 

74°    0'    3"  W 

Ann  Arbor  

80°  43'         W 

New  Orleans 

90°    2'  30"  W 

Boston  

71°    3'  30"  W. 

Paris  

....     2°  20'  22"  E 

Berlin  

13°  23'  45"    E 

Philadelphia 

75°  107         W 

Calcutta  

88°  19'    2"    E. 

Rome 

12°  27'  14"   E 

84°  29'  31"  W 

Richmond  "Va 

77°  25'  45"  W 

87°  37'  45"  W 

San  Francisco 

122°  26'  45"  W 

Jefferson  City,  Mo.  .  . 

92°    8'         W. 

St.  Paul,  Minn. 

.  95°    4'  55"  W. 

London  

0°    5'  38"  W 

St  Louis   Mo 

90°  15'  15"  W 

Mexico.  .  . 

99°    5'         W. 

Washington.  D. 

C..    .77°    &  15"  W. 

352.  Standard  Time.— In  1883,  the  principal  railroads  and  cities 
of  the  United  States  and  Canada  adopted  the  time  of  four  different 
meridians  as  the  standard  time  of  four  belts  or  sections  comprising  the 
whole  of  the  above  countries.  The  most  eastern  of  these  sections  embraces 
the  Eastern  and  Middle  States,  Maryland  and  Virginia,  and  extends  about 
7£°  east  and  west  of  the  meridian  of  75°  west  of  Greenwich  (near  Philadel- 
phia). The  time  of  this  meridian,  called  Eastern  time,  is  used  in  this  section, 
and  is  5  hours  slower  than  Greenwich  time.  The  time  of  the  meridian  of 
90°  west  of  Greenwich  (near  St.  Louis)  called  Central  time,  is  used  in  the  next 
section,  and  is  1  hour  slower  than  Eastern  time.  The  time  of  the  meridian 
of  105°  west  of  Greenwich  (near  Denver),  called  Mountain  time,  is  used  in 
the  Rocky  Mountain  region,  and  is  1  hour  slower  than  Central  time  and 
2  hours  slower  than  Eastern  time.  The  time  of  the  meridian  of  120°  west  of 
Greenwich,  called  Pacific  time,  is  used  in  the  Pacific  slope,  and  is  3  hours 
slower  than  Eastern  time. 


Art.  353.]  LONGITUDE     AND      TIME.  127 

EXAMPLES. 

353.  Find  the  difference  in  longitude  between 

1.  New  York  and  London.         4.  -St.  Louis  and  Calcutta. 

2.  Boston  and  Paris.  5.  Philadelphia  and  Berlin. 

3.  Chicago  and  San  Francisco.   6.  San  Francisco  and  Calcutta. 

Find  the  difference  in  solar  time  between 

7.  New  York  and  Greenwich.   10.  Rome  and  London. 

8.  Chicago  and  New  York.        11.  Paris  and  Albany. 

9.  Richmond  and  Calcutta.       12.  Calcutta  and  Jefferson  City. 

Find  mentally  the  difference  in  standard  time  between 

18.  Albany  and  Denver.  16.  St.  Louis  and  Richmond. 

14.  New  York  and  Chicago.     17.  St.  Paul  and  Sacramento. 

15.  Boston  and  San  Francisco.  18.  Phila.  and  Portland,  Me. 

Find  the  difference  between  the  standard  time  and  the  solar 
time  of  the  following  cities: 

19.  Boston.  21.  San  Francisco.    28.   Chicago. 

20.  Philadelphia.    22.  St.  Louis.  24.  St.  Paul. 

25.  A  navigator  finds  that  when  it  is  noon  at  his  place  of 
observation,  it  is  16  min.  34  sec.  past  10  P.M.  by  his  chronometer, 
Greenwich  time  ;  what  is  his  longitude  ? 

26.  When  it  is  6  o'clock  P.M.,  standard  time,  at  Richmond, 
Va.,  what  is  the  time  at  St.  Louis,  Mo.? 

27.  If  the  difference  of  solar  time  between  two  places  is  1  hr. 
18  min.  4  sec.,  what  is  the  difference  of  longitude  ? 

28.  When  it  is  20  min.  past  2  P.M.,  standard  time,  at  Boston, 
Mass.,  what  o'clock  is  it  at  San  Francisco  ? 

29.  When  it  is  Monday,  10  P.M.,  standard  time,  in  Chicago, 
what  day  and  time  is  it  in  London  (Greenwich  time)  ? 

30.  When  it  is  9  o'clock  P.M.,  solar  time,  in  San  Francisco,  it 
is  3  min.  3T2^  sec.  past  11  A.M.  in  Calcutta  ;  what  is  the  longitude 
of  San  Francisco,  if  the  longitude  of  Calcutta  is  88°  19'  2"  E.  ? 

81.  When  it  is  noon,  solar  time,  in  Chicago,  it  is  5  min.  29-J- 
sec.  of  1  P.M.,  solar  time,  in  New  York  ;  what  is  the  longitude 
of  Chicago,  the  longitude  of  New  York  being  74°  3"  W.  ? 


128  DENOMINATE     NUMBERS.  [Art.  354. 


THE   METRIC   SYSTEM.* 

354.  In  the  Metric  System  of  weights  and  measures,  the 
Meter  is  the  basis  of  all  the  units  which  it  employs. 

355.  The  Meter  is  the  unit  of  length,  and  is. equal  to  one 
ten-millionth  part  of  the  distance  measured  on  a  meridian  of  the 
earth  from  the  equator  to  the  pole,  and  equals  about  39.37  inches. 

The, standard  meter  is  a  bar  of  platinum  carefully  preserved  at  Paris. 
Exact  copies  of  the  meter  and  the  other  units  have  been  procured  by  the 
several  nations,  including  the  United  States,  that  have  legalized  the  system. 
Comparisons  with  the  standard  units  are  made  under  certain  conditions  of 
temperature  and  atmospheric  pressure. 

356.  The  names  of  the  higher  denominations,  or  multiples, 
of  the  unit  are  formed  by  prefixing  to  the  several  units  the  Greek 
numerals,  deka  (10),  hecto  (100),  Mo  (1000),  and  myria  (10000); 
as  dekameter,  10  meters,  hectometer,  100  meters,  etc. 

To  assist  the  memory,  observe  that  the  initial  letters  of  the  multiples  are 
in  alphabetical  order  ;  thus,  D,  H,  K,  and  M. 

357.  The  names  of  the  loiver  denominations,  or  divisions,  of 
the  unit  are  formed  by  prefixing  to  the  several  units  the  Latin 
numerals,  deci  (-fa),  centi  (y^),  milli  ( i  0*0  0 ) ;  as  decimeter,  fa 
meter,  centimeter,  y^  meter,  etc. 

To  assist  the  memory  observe  that  the  following  words  are  derived  from 
the  same  roots :  dime,  decade,  decimal,  decimate,  decennial,  etc. ;  cent,  cental, 
century,  centennial,  etc.;  mill,  millennium,  etc. 

LINEAR     MEASURE. 

358.  TABLE. 


10mm. 
10  cm. 
10  dm 

= 

1 
1 
1 
1 

Millimeter.  .  . 
Centimeter.  .  . 
Decimeter.  .  .  . 
METER  .  ... 

••(unnr  of  a  meter) 
.  .  (x^  of  a  meter) 
..(^  of  a  meter) 
(1  meter) 

= 

.03937  in. 
.3937  in. 
3.937  in. 
39  37  in. 

10  m. 

_ 

1 

Dekameter.  .  . 

.  .(10  meters) 

_ 

32.8ft. 

10  Dm. 
10  Hm. 

= 

1 

1 

Hectometer  .  . 
Kilometer  

..(100  meters) 
..(1000  meters) 

— 

328.09ft. 
.62137  mi. 

For  other  foreign  weights  and  measures,  see  page  343. 


Art.  358.]  THE    METRIC    SYSTEM.  129 

NOTES. — 1.  The  meter,  like  the  yard,  is  used  in  measuring  cloths,  ribbons, 
laces,  short  distances,  etc. 

2.  The  kilometer  is  used  in  measuring  long  distances,  and  is  about  f  of  a 
mile. 

3.  The   centimeter  and   millimeter  are  use  by  artisans  and   others  in 
measuring  minute  lengths.     The  other  denominations  are  rarely  used. 

EXAMPLES. 

359.  1.  Eeduce  875275  meters  to  kilometers. 

ANALYSIS. — Since  1  kilometer  equals  1000  meters,  in  875275  meters  there 
are  as  many  kilometers  as  1000  is  contained  times  in  875275,  or  875.275.  To 
divide  by  1000,  place  the  point  three  places  to  the  left  (27O,  4). 

2.  Eeduce  675.318  kilometers  to  meters. 

ANALYSIS. — Since  1  kilometer  equals  1000  meters,  in  675.318  kilometers 
there  are  675.318  times  1000,  or  675318  meters.  To  multiply  by  1000,  place 
the  point  three  places  to  the  right  (267,  note). 

3.  Reduce  383.64  meters  to  centimeters  ;  to  kilometers. 

4.  Reduce  175.16  centimeters  to  kilometers  ;  -to  meters. 

5.  Reduce  to  meters  and  find  the  sum  of  876.2   decimeters, 
30347  centimeters,  176.48  meters,  8.175  kilometers. 

6.  A  ship  sails  5712  kilometers  in  48  days ;  how  many  kilo- 
meters does  she  sail  per  day  ? 

7.  What  is  the  value  of  56.4  meters  of  silk  at  $1.75  per  meter? 

8.  16  pieces  of  cloth  contain  38.5  meters  each  ;  18  pieces  con- 
tain 39  meters  each;  and  24  pieces  contain  41.2   meters   each; 
how  many  meters  in  all? 

9.  How  many  meters  of  ribbon  at  27  cents  per  meter  can  be 
purchased  for  $245. 70  ? 

10.  If  the  forward  wheels  of  a  carriage  are  3.5  meters  in  cir- 
cumference, and  the  hind  wheels  4.8  meters,  how  many  more 
times  will  the  forward  wheels  revolve  than  the  hind  wheels,  in 
running  a  distance  of  8.4  kilometers  ? 

11.  How  much  will  it  cost  to  sewer  a  street  .64  Km.  long,  at 
$3.75  per  meter? 

12.  How  many  meters  of  wire  will  be  required  to  fence  a 
rectangular  field,  .72  Km.  long  and  .56  Km.  wide,  if  the  fence  is 
4  wires  high  ? 

13.  How  long  will  it  take  a  railway  train,  running  60  Km. 
per  hour,  to  go  from  New  York  to  Chicago,  the  distance  being 
1440  Km.? 


130  DENOMINATE     NUMBERS.  [Art.  36O 

SQUARE     MEASURE. 

360.  The  unit  of  square  measure  is  the  square  meter. 

TABLE. 

100  Square  Centimeters,  sq.  cm.  =  1  Square  Decimeter  =  15.5+  sq.in. 

100  Square  Decimeters,  sq.  dm.  =  1  SQUARE  METER,  Sq.  M.  —  1.196+  sq.  yd. 

NOTES. — 1.  The  square  meter  is  used  in  measuring  flooring,  ceilings,  etc. ; 
the  square  decimeter  and  the  square  centimeter  are  used  for  minute  surfaces. 

2.  Since  units  of  square  measure  form  a  scale  of  hundreds,  each  denomi- 
nation must  have  two  places  of  figures. 

361.  The  unit  of  Land  Measure  is  the  are,  and  is  equal  to 
a  square  dekameter  (100  square  meters),  or  119.6  square  yards. 

TABLE. 

1  Centare ....  (1  square  meter)  =  1550  sq.  in. 

100  Centares,  ca.  •=   1  Are (100  square  meters)       =  119.6  sq.  yd. 

100  Ares,          A.  =   1  Hectare. . .  .(10000  square  meters)  =  2.471  acres. 

NOTE. — The  hectare  is  the  ordinary  unit  for  land. 
EXAMPLES. 

362.  1.  Write  16  sq.  m.,  8  sq.  dm.,  24  sq.  cm.,  having  the 
square  meter  as  the  unit.     (36O,  2.)  Ans.  16.0824. 

2.  Write  83  sq.  m.,  9  sq.  dm.,  having  the  sq.  m.  as  the  unit. 
8.  In  47  ares  how  many  square  meters  ? 

4.  In  60.25  hectares  how  many  centares  ? 

5.  How  many  square  meters  in  a  building  lot  8  m.  by  32  m.  ? 

6.  How  many  building  lots,  each  containing  225  sq.  m.,  can  be 
formed  from  a  field  containing  9  hectares  ? 

7.  How  many  hectares  in  a  farm  1.024  Km.  in  width  and 
1.625  Km.  in  length? 

8.  What  is  the  cost  of  a  mirror  2.25  m.  by  1.44  m.,  at  $3.84 
per  sq.  m.  ? 

9.  How  many  lots  25  m.  wide  and  60  m.  deep,  or  having  an 
equivalent  area,  can  be  laid  out  from  6  hectares  ? 

10.  A  man  bought  a  piece  of  land  for  $6950.50,  and  sold  it  for 
$7603.30,  by  which  transaction  he  made  $6.80  a  hectare;  how 
many  hectares  were  there  ? 


Art.  363.]  THE     METRIC     SYSTEM.  131 


CUBIC     MEASURE. 

363.  The  unit  for  measuring  ordinary  solids   is   the  cubic 
meter. 

TABLE. 

1000  Cu.  Millimeters,  cu.  mm.  =  1  Cu.  Centimeter  =     .061  cu.  in. 
1000  Cu.  Centimeters,  cu.  cm.  =  1  Cu.  Decimeter     =     61.027  cu.  in. 

1000  Cu.  Decimeters,   cu.  dm.  =  1  Cu.  METER          =  (  ?5'^7  CU'  &' 

\  1.308  cu.  yd. 

NOTES. — 1.  The  cubic  meter  is  used  in  measuring  embankments,  excava- 
tions, etc. ;  cubic  centimeters  and  cubic  millimeters  for  minute  bodies. 

2.  Since  units  of  cubic  measure  form  a  scale  of  thousands,  each  denomi- 
nation must  have  three  places  of  figures. 

364.  The  unit  of  "Wood  Measure  is  the  ster,  and  is  equal 
to  a  cubic  meter,  or  35.317  cubic  feet. 

TABLE. 

10  Decisters,  ds.  =  I  Ster. .  .  .(1  Cubic  Meter)     =  (  l^fj50*1' 

{.  OO.ol/  CU.  jt. 

10  Sters,  s.  =  1  Dekaster,  Ds.  .(10  Cubic  Meters)  =     2.759  cords. 


EXAMPLES. 

365.     1.  Write  29  cu.  m.,  75  cu.  dm.,  having  the  cubic  meter 
as  the  unit.     (363,  2)  Ans.  29.075  cu.  m. 

2.  Write  17  cu.  m.,  218  cu.  dm.,  27  cu.  cm.,  having  the  cubic 
meter  as  the  unit. 

3.  How  many  cubic  meters  in  a  box  3.5  m.  by  3.2  m.  by  2.5  m.  ? 

4.  Bought  12  sters  of  wood  ;  having  sold  8. 7  cubic  meters, 
how  much  remained  ? 

5.  There  are  13  blocks  of  marble,  each  containing  370.16  cu. 
dm.  •  how  many  cubic  meters  in  all  ? 

6.  How  many  cubic  meters  in  an  excavation  13.2  m.  by  18.5  m. 
by  8.4  m.? 

7.  At  $1,25  a  cubic  meter,  what  will  it  cost  to  dig  a  cellar 
6.5  m.  long,  5.4  m.  wide,  and  2.5  m.  deep  ? 

8.  How  many  sters  of  wood  in  a  pile  of  wood  2.5  m.  high, 
2  m.  wide,  and  16.5  m.  long  ?     What  is  the  length  of  a  pile  of  the 
same  height  and  width  containing  216  sters  ? 


DENO  Ml  NA  TE     3r  UKB  ERS.  [Ar» .  3C6. 


DRY     AND     LIQUID     MEASURE. 

366.  The  unit  of  Dry  and  Liquid  Measure  is  the  liter, 
which  is  equal  to  a  cubic  decimeter,  1.0567  wine  quarts,  or  .908 
dry  quart. 

TABLE. 

Dry  Measure.  Liquid  Measure. 

1  Milliliter (y^  of  a  liter)  =  .06103  cu.  in.,  or,  .0338  yJ.  oz. 

10ml.    =  1  Centiliter (^  of  a  liter)  —  .6103  cu.  in.,  or,  .338^.0,2. 

10  el.     —  1  Deciliter (TV  of  a  liter)  =  6.1027  cu.  in.,  or,  .845  gi. 

10  dl.     =  1  LITER (1  liter)  =  .908  qt.,  or,  1.0567  qt. 

10  I.       =±  1  Dekaliter (10  liters)  =  9.08  qt.,  or,  2.6418  gal. 

10  Dl.   =  1  Hectoliter  . . .  .(100  liters)  =  2.8375  bu.,  or,  26.418  gal. 

10  HI.  =  1  Kiloliter (1000  liters)  =  28.375  bu.,  or,  264.18^. 

NOTES. — 1.  The  liter  is  commonly  used  in  measuring  wine,  milk,  etc.,  in 
moderate  quantities.  For  minute  quantities  the  centiliter  and  milliliter  are 
employed  ;  and  for  large  quantities  the  dekaliter. 

2.  For  measuring  grain,  etc.,  the  hectoliter  (2.8375  bushels)  is  commonly 
used. 

3.  Instead  of  the  kiloliter  and  milliliter,   it  is  customary  to  use  their 
equals,  the  cubic  meter  and  cubic  centimeter. 

EXAM  PLES. 

367.  1.  How  many  liters  in  a  vessel  whose  capacity  is  1 
cubic  meter  ? 

2.  What  is  the  cost  of  sixteen  liters  of  milk  at  8  cents  a  liter  ? 

3.  How  many  hectoliters  of  wheat  can  be  bought  for  $396  at 
$5.50  per  hectoliter  ? 

4.  How  many  hectoliters  of  grain  can  be  put  in  a  rectangular 
bin,  4  m.  long,  3.5  m.  wide,  and  1.2  m.  high  ? 

5.  How  many  liters  in  63.5  dekaliters  ?     In  83.75  hectoliters  ? 

6.  At  $1.75  a  liter,  what  is  the  cost  of  85.6  dekaliters  of  wine? 

7.  How  many  hectoliters  in  16  cubic  meters  ? 

8.  How  many  bags,  each  holding  1  hectoliter,  can  be  filled 
from  a  bin,  1.5  m.  high,  2.4  m.  wide,  and  5  m.  long  ? 

9.  A  cistern  3.5  m.  by  3.2m.,  and  9  m.  deep,  will  hold  how 
many  dekaliters  ? 

10.  A  merchant  bought  4  hectoliters  of  nuts  at  $8.50  per 
hectoliter,  and  retailed  them  at  12  cents  a  liter ;  what  was  his 
profit  ? 


Art.  368.]  THE     METRIC     SYSTEM.  133 


WEIGHT. 

368.  The  unit  of  weight  is  the  gram,  which  is  equal  to  the 
weight  of  a  cubic  centimeter  of  distilled  water  in  a  vacuum;  at  its 
greatest  density  (39.2°  F.),  or  15.432  grains. 

TABLE. 

1  Decigram (-^  of  a  gram)  =     1.543  gr.  Tr. 

10  dg.  =      1  GRAM (1  gram)  =     15.432  gr.  Tr. 

10  g.  =     1  Dekagram (10  grams)  =     .3527  oz.  Av. 

10  Dg.  =     1  Hectogram (100  grams)  =     3.5274  oz.  Av. 

10  Hg.  =     1  Kilogram (1000  grams)  =     2.2046  Ib.  Av. 

10  Kg.  —     1  Myriagram (10000  grams)  =     22.040  Ib.  Av. 

100  Kilos       =     1  Quintal. . (100000  grams)  =     220.46  Ib.  Av. 

10  Q.,  or     \      /  1  Tonneau,  )  (  2204.6  Ib.  Av. 

1000  Kilos /  =  (     or  TON      \-^      )0  grams)  :=  j  1.1023  T. 

NOTES. — 1.  The  above  table  is  used  in  computing  the  weights  of  all 
objects  from  the  smallest  atom  to  the  largest  known  body.  The  gram,  kilo- 
gram (or  kilo),  and  ton  are  principally  used. 

2.  The  gram  is  used  in  weighing  letters,  gold,  silver,  and  medicines. 

3.  The  kilogram,  or  kilo,  like  the  pound,  is  used  in  weighing  groceries 
and  coarse  articles.     It  is  approximately  2i  pounds  Av. 

4.  The  ton  is  the  weight  of  a  cubic  meter  of  water,  and  is  used  in  weigh- 
ing very  heavy  articles,  as  coal,  iron,  etc. 

5.  The  pound  of  Germany,  Austria,  and  Denmark  is  equal  to  £  of  a  kilo- 
gram ;  the  centner,  to  100  pounds,  or  |  of  a  quintal. 

EXAMPLES. 

369.  1.  What  is  the  weight  in  grams  of  a  cubic  meter*  of 
water  ?     Of  a  cu.  dm.  of  water  ? 

2.  A  farmer  sells  to  A  3.716  T.  of  hay,  to  B  4.325  T.,  to  C  8775 
kilos  ;  how  many  tons  does  he  sell  ? 

3.  The  U.  S.  50-cent  piece  weighs  12.5  grams  ;  how  many  can 
be  coined  from  a  kilogram  of  standard  silver  ? 

4.  The  U.  S.  5-cent  piece  weighs  5  grams  ;  how  many  5-cent 
pieces  are  equivalent  in  weight  to  12  50-cent  pieces  ? 

5.  How  much  alloy  must  be  used  in  making  1200  U.  S.  twenty- 
five-cent  pieces  ?     (See  Art.  113.) 

6.  What  is  the  cost  of  75.6  kilos  of  sugar  at  18  cents  a  kilo  ? 

7.  How  many  powders,  each  containing  6  grams,  can  be  made 
from  .372  kilogram  ? 

8.  What  is  the  weight  of  10  cii.  m.  of  ice,  it  being  .93  as 
heavy  as  water  ? 


134  DENOMINATE     NUMBERS.  [Art.  37 O. 


37O.  TABLE  OF  EQUIVALENTS. 

1  inch  =  2.54  centimeters 1  centimeter  =  0.3937  inch. 

1  foot  =  3.048  decimeters 1  decimeter  =  0.328  foot. 

1  yard  =  0.9144  meter 1  meter  =  1.0936  yards  =  39.37  in. 

1  rod  =  0.5029  dekameter 1  dekameter  =  1.9884  rods. 

1  mile  =  1.6093  kilometers 1  kilometer  =  0.62137  mile. 

1  sq.  inch  =  6.452  sq.  centimeters 1  sq.  centimeter  =  0.155  sq.  inch. 

1  sq.  foot  =  9.2903  sq.  decimeters 1  sq.  decimeter  =  0.1076  sq.  foot. 

sq.  yard  =  0.8361  sq.  meter 1  sq.  meter  =  1.196  sq.  yards. 

sq.  rod  —  25.293  sq.  meters 1  are  =  3.954  sq.  rods  =  119.6  sq.  yards. 

acre  =  0.4047  hectare 1  hectare  =  2.471  acres. 

sq.  mile  =  2.59  sq.  kilometers 1  sq.  kilometer  —  0.3861  sq.  mile. 

cu.  inch  =  16.387  cu.  centimeters. ..  1  cu.  centimeter  =  0.061  cu.  inch. 

cu.  foot  =  28.317  cu.  decimeters 1  cu.  decimeter  =  0.0353  cu.  foot, 

cu.  yard  =  0.7645  cu.  meter 1  cu.  meter  =  1.308  cu.  yards. 

1  cord  =  3.624  sters 1  ster  =  0.2759  cord. 

1  liquid  quart  =  0.9463  liter 1  liter  =  1.0567  liquid  quarts. 

1  gallon  =  0.3785  dekaliter 1  dekaliter  =  2.6417  gallons. 

1  dry  quart  =  1.101  liters 1  liter  =  0.908  dry  quart. 

1  peck  =  0.881  dekaliter 1  dekaliter  =  1.135  pecks. 

1  bushel  =  3.524  dekaliters 1  hektoliter  =  2.8375  bushels. 

1  ounce  av.  =  28.35  grams 1  gram  =  0.03527  ounce  av. 

1  pound  av.  =  0.4536  kilogram 1  kilogram  =  2.2046  pounds  av. 

1  pound  av.  =  0.9072  German  pounds.  1  German  pound  =  1.1023  pounds  av. 
1  ton  (2000  Ibs.)  =  0.9072  met.  ton. ...  1  met.  ton  =  1.1023  tons  =  2204.6  Ib.  av. 

1  grain  Troy  =  0.0648  gram 1  gram  =  15.432  grains  Troy. 

1  ounce  Troy  =  31.1035  grams 1  gram  =  0.03215  ounce  Troy. 

1  ppund  Troy  =  0.3732  kilogram 1  kilogram  =  2.679  pounds  Troy. 


EXAMPLES. 

371.    1.    In    226    meters    how   many   yards  ?      How   many 

inches  ? 

2.  Reduce  G  miles  to  kilometers  ;  to  meters. 
8.  Eeduce  640  acres  to  hectares  ;  to  ares. 

4.  In  J.O  kilometers,  how  many  feet  ?     How  many  miles  ? 

5.  In  375.6  kilos,  how  many  pounds  ? 

6.  How  many  German  pounds  in  225  English  or  U.  S.  pounds? 

7.  What  is  the  weight  of  the  U.  S.  standard  silver  dollar  in 
grams  ?     Of  the  trade  dollar  ? 

8.  In  5000  U.  S.  bushels,  how  many  hectoliters  ?    How  many 
dekaliters  ? 

9.  In  875  cu.  yd.  how  many  cu.  m.  ? 


Art.  371.] 


APPROXIMATE    RULES. 


135 


10.  In  1000  cu.  m.  how  many  cu.  yd.  ? 

11.  Reduce  1728  gal.  wine  to  liters  ;  to  dekaliters. 

12.  In  244  sq.  m.  how  many  sq.  yd.  ?     How  many  sq.  ft.  ? 
IS.  Reduce  220  oz.  Av.  to  grams  ;  to  kilograms. 

372.    APPKOXIMATE  VALUES. 

When  no  great  accuracy  is  required,  we  may  consider — 


1  decimeter       =  4  inches. 

1  meter  =  39  inches. 

5  meter's  —  1  rod. 

1  kilometer       =  f  mile. 

1  square  meter  =  lOf  square  feet. 

1  hectare  =  24  acres. 


1  cu.  met.  or  ster  =  1£  cu.  yd.  or  \  cord. 


1  liter 
1  hectoliter 
1  gram 
1  kilogram 
1  ton 


=  1  quart. 
=  2^  bushels. 
=  15^  grains. 
=  2£  pounds. 
=  2200  pounds. 


APPROXIMATE     RULES. 

373.  To  reduce  avoirdupois  pounds  to  kilograms : 

Divide  by  2,  and  then  deduct  one-tenth. 

NOTE. — Answer  too  small  by  about  8  kilos  for  every  1000  kilos  of  the 
result.  If  -jJj-,  instead  of  TV,  be  deducted,  the  answer  will  be  too  great  by  2 
kilos  for  every  1000  kilos  of  the  result. 

374.  To  reduce  avoirdupois  pounds  to  half-kilograms, 
or  German  pounds : 

Deduct  one-tenth. 

NOTE. — Answer  too  small  by  about  8  German  pounds  for  every  1000  Ger- 
man pounds  of  the  result.  If  ^  be  deducted,  the  answer  will  be  too  great  by 
2  German  pounds  for  every  1000  German  pounds  of  the  result. 

375.  To  reduce  yards  to  meters: 

Deduct  one-twelfth. 

NOTE. — Answer  too  great  by  2£  m.  for  every  1000  m.  of  the  result. 

376.  To  reduce  square  yards  to  square  meters : 

Deduct  one-sixth. 

NOTE. — Answer  too  small  by  about  3  sq.  m.  for  every  1000  sq.  in.  of  the 
result. 

377.  To  reduce  cubic  yards  to  cubic  meters  : 

Divide  by  1.3. 

NOTE. — Answer  too  great  by  about  6  cu.  m.  for  every  1000  cu.  m.  of  the 
result. 


136  DENOMINATE     NUMBERS.  [Art.  378. 

378.  To  reduce  U.  S.  gallons  to  liters  : 

Multiply  by  h  and  then  subtract  one-twentieth  (5 per  cent.). 
NOTE. — Answer  too  great  by  about  4  /.  for  every  1000  I.  of  the  result. 

379.  To  reduce  U.  S.  bushels  to  hectoliters : 

Divide  by  8,  and  then  add  one-twentieth  (5 per  cent.). 

NOTE. — Answer  too  small  by  about  7  hi.  for  every  1000  hi.  of  the  result. 

380.  To  reduce  kilograms  to  avoirdupois  pounds : 

Multiply  by  2,  and  then  add  one-tenth. 

NOTE.— Answer  too  small  by  about  2  Ib.  for  every  1000  Ib.  of  the  result. 

381.  To  reduce  German  pounds,  or  half-kilograms, 
to  avoirdupois  pounds : 

Add  one-tenth. 

NOTE. — Same  error  as  in  Art.  38O. 

382.  To  reduce  meters  to  yards : 

Add  one-twelfth  and  1%  'of  the  original  number. 

NOTE. — Answer  too  small  by  only  \  yd.  for  every  1000  yd.  of  the  result. 

Dealers  in  dry  goods  add  only  T^  in  reducing  meters  to  yards,  and  thus 
make  the  result  too  small  by  about  9£  yd.  for  every  1000  yd.  of  the  result. 

If  -fa  be  added,  the  answer  will  be  too  small  by  about  2£  yd.  for  every  1000 
yd.  of  the  result.  If  y1^  be  added,  the  answer  will  be  too  great  by  about  6  yd. 
for  every  1000  yd.  of  the  result. 

383.  To  reduce  square  meters  to  square  yards  : 

Add  one-fifth. 

NOTE. — Answer  too  great  by  about  3  sq.  yd.  for  every  1000  sq.  yd.  of  the 
result. 

384.  To  reduce  cubic  meters  to  cubic  yards : 

Multiply  by  1.3. 

NOTE. — Answer  too  small  by  about  6  cu.  yd.  for  every  1000  cu.  yd.  of  the 
result. 

385.  To  reduce  liters  to  U.  S.  gallons : 

Multiply  by  2. 11,  and  then  divide  by  8. 

NOTE. — Answer  too  small  by  about  1.7  gal.  for  every  1000  gal.  of  the  result. 

386.  To  reduce  hectoliters  to  U.  S.  bushels  : 

Multiply  by  3,  and  then  subtract  one-twentieth  (5 per  cent.).    . 
NOTE. — Answer  too  great  by  about  4  bu.  for  every  1000  bu.  of  the  result. 


Art.  387.]  REVIEW     EXAMPLES.  137 


REVIEW     EXAMPLES. 

387.  1.  Add  174,  26J,  35f,  4S£,  and  8TV  ;  multiply  the  sum 
by  59  ;  subtract  2309T\  from  the  product ;  and  divide  the  remain- 
der by  162f . 

2.  Divide  fourteen,  and  twenty-five  hundredths  by  one  hun- 
dred twenty-five  thousandths  ;  add  nineteen,  and  sixty-four  hun- 
dredths to  the  quotient ;  and  multiply  the  sum  by  eight,  and  five 
tenths. 

3.  Find  the  time  by  Compound  Subtraction  and  in  exact  days 
from  March  24  to  Sept.  18. 

4.  How  many  lengths  of  pipe,  each  10  ft.  (')  3  in.  (")  long, 
will  be  required  for  a  well  130  ft.  deep  ? 

5.  A  horse  trots  a  mile  in  2  min.  45  sec.     How  many  feet  is 
that  per  second  ? 

6.  A  grass  plot  13  ft.  by  54  ft.  is  surrounded  by  a  stone  walk 
\\ft.  wide.     The  stone  walk  is  surrounded  by  a  gravel  road  7J/rf. 
wide.     How  many  square  feet  are  covered  by  the  grass,  the  stone, 
and  the  gravel  respectively  ?     (Make  diagram.) 

7.  What  is  the  cost  of  15669  pounds  meal  at  $1.10  per  cwt.  ? 

8.  What  is  the  cost  of  16450  pounds  of  hay  at  $15.50  per  ton? 

9.  How  many  square  rods  in  a  triangular  piece  of  land,  360 
rd.  long  and  whose  perpendicular  width  is  240  rd.  ? 

NOTE. — To  find  the  area  of  a  triangle,  take  one-half  the  product  of  the 
length  (base)  and  the  height  or  width  (altitude). 

10.  How  many  square  feet  in  the  gable  of  a  house,  40  ft.  long 
and  24:  ft.  high  ? 

11.  How  many  feet  of  siding  would   be  required  for  a  house 
40  ft.  long,  24  ft.  wide,  IS  ft.  high,  with  two  gables  each  24  ft. 
wide  and  12  ft.  high,  adding  one-fifth  for  the  lap  and  waste  in 
cutting  ? 

12.  How  much  will  it  cost  to  make  an  excavation,  40  ft.  long, 
30  ft.  wide,  and  9ft.  deep,  at  32c.  per  cubic  yard. 

13.  How  many  feet  of  2-inch  plank,  making  no  deduction  for 
the  corners,  would  be  needed  to  build  a  rectangular  tank,  without 
a  cover,  10  ft.  long,  6$  ft.  deep,  Sft.  wide  ? 

14>  The  circumference  of  any  circle  is  equal  to  the  diameter 
multiplied  by  3.1416  (about  3^).  Find  the  circumference  of  a 
circle,  whose  diameter  is  5  feet. 


138  DENOMINATE     NUMBERS.  [Art.  387. 

"^15.  The  area  of  any  circle  equals  the  square  of  the  radius 
multiplied  by  3.141G  (3|),  or  the  square  of  the  diameter  multi- 
plied by  .  7854.  What  is  the  area  of  a  circle  whose  diameter  is 
6  ft.  ?  Whose  radius  is  5  ft.  ? 

16.  How  many  feet  of  2-inch  plank  would  be  required  to  make 
a  cylindrical  cistern  without  a  cover,  7  ft.  in  diameter  and  8  ft. 
high? 

17.  How  many  pounds  in  16  T.  3  qr.  18  Ib.  (Long  Ton  Table)? 

18.  How  many  quarts  in  3  Vbl.  24  gal.  cider  ? 

19.  In  27318  pounds  of  corn,  how  many  bushels  ?     What  is 
the  value  of  the  same  at  48f  cents  per  bushel  ? 

^  W.  What  is  the  value  of  27318  pounds  of  corn,  at  87.1  cents 
per  cental  ? 

NOTE. — Examples  19  and  20  illustrate  the  present  and  the  centai  systems 
of  buying  and  selling  produce,  and  show  the  calculations  saved  by  using  the 
latter. 

21.  Paid  $222.75  for  boards  at  $13.50  per  M.;  how  many  feet 
were  purchased  ? 

22.  What  is  the  value  of  27315  ft.  of  lumber  at  $12  per  M.  ? 

23.  A  quartermaster  purchased  75000  pounds  of  corn,  at  31J- 
cents  per  bushel ;  32113  pounds  of  oats,  at  32]  cents  per  bushel ; 
and  79500  pounds  of  hay,   at  $22. 37^  per  ton  (2000  pounds). 
"What  was  the  total  cost  of  the  purchase  ? 

24.  A  farmer  sold  18360  pounds  of  corn,  at  64  cents  per  cen- 
tal ;  22450  pounds  of  oats,  at  94  cents  per  cental ;  and  36650 
pounds  of  hay,  at  $1.31  per  cental.     How  much  was  realized  from 
the  sale  ? 

25.  Reduce  £19  16s.  9d.  to  the  decimal  of  a  pound. 

26.  If  £1  sterling  is  worth  $4.87,  what  is  the  value  of  £225 
185.  Qd.  ? 

27.  From  £16  12s.  9rf.  deduct  .05  of  itself. 

28.  What  is  the  value  of  20  yd.  silk  at  10s.  Gd.  per  yard  ? 

29.  The  difference  in  the  local  time  of  two  places  is  3  Ur. 
43  min.  12  sec. ;  what  is  the  difference  in  longitude  ? 

30.  What  is  the  capacity  in  liters  of  a  cistern  25  meters  long, 
2.2  meters  wide,  and  3  meters  deep  ? 

81.  The  specific  duty  on  Brussels  carpet  is  44  cents  per  square 
yard  ;  what  is  the  duty  per  square  meter  ? 

32.  The  duty  on  tallow  candles  is  2£  cents  per  pound  ;  what 
|  is  the  duty  per  kilogram  ? 


ALIQUOT    PARTS. 


388.  An  Aliquot  Part  of  a  number  or  quantity  is  a  number 
that  will  divide  it  without  a  remainder  ;  as  20  of  60,  12£  of  100, 
4  of  12,  etc. 

Any  fraction  having  1  as  its  numerator  is  an  aliquot  part  of  a  unit. 
Many  of  the  ordinary  business  computations  can  be  shortened  by  the  use 
of  aliquot  parts. 

EXAMPLES. 

389.     1.  Find  the  cost  of  217  pounds  of  sugar  at  8£<?.  per 
pound?     AtSfe.?    At7fc.? 

OPERATION. 


.08J- 
1736 
109  ( 
54  { 

18.99 


NOTE.—  f  = 


=  I  of 


Find  the  cost  of 

2.  |  doz.  elbows  at  $2.75  per  dozen. 

3.  141  Ibs.  raisins  at  8|c.  per  pound. 

4.  200  Ibs.  lead  at  6f  c.  per  pound. 

5.  228  Ibs.  putty  at  2|c.  per  pound. 

6.  207  Ibs.  currants  at  5Jc.  per  pound. 

7.  102^  (J  +  -J-)/*.  tubing  at  16c.  per  foot. 

8.  877  /6s.  paper  at  3|c.  per  pound. 
P.  102  Ibs.  oatmeal  at  3|c.  per  pound. 

10.  700  /6s.  soap  at  5f  c.  per  pound. 

11.  503  /&>'.  ham  at  10}c.  per  pound. 

12.  644  Ibs.  lard  at  lljc.  per  pound. 

13.  2957  /6s.  sugar  at  S-foc.  (J-f  J  of  J)  per  pound. 


140  ALIQUOT    PARTS.  [\rt.390. 

39O.  Aliquot  parts  of  100. 


3*  =  A-  16f  =  *•  62i  =  i  +  i  (i  of  1). 

4  =A-  20    =  t.  75    =J  +  i(*ofi). 

5  =  ,V-  25    -  i-  S?i  =  *  +  i  +  *- 

6J  -  A,  33i  =  J.  18|  =  i  +  A  (*  of  1). 

10    =      .  50    =  31     =  of      . 


NOTE.  —  In  the  following  commercial  problems,  use  as  few  figures  as 
possible. 

EXAMPLES. 

391.     1.  Find  the  cost  of  13756  pounds  of  meal  at  $1.05 
per  cwt. 

OPERATION. 

137.56  ANALYSIS.  —  At  $1  per  cwt.  the  cost  would  be  $137.56. 

g  gg        5c.  =  fa  of  $1.    To  divide  by  20,  divide  by  2  and  place  the 

quotient  figures  one  place  to  the  right. 
144.44 

2.  Find   the  value  of  16345  Ibs.   of  feed  at    $1.10  per  cwt. 
(We  =  A  of  *!)• 

3.  What  is  the  cost  of  12  doz.  hats  at  $4.12£  (£)  per  doz.  ? 

4.  Find  the  cost  of  471f#  (£  +  £)  bushels  of  corn  at  41c. 

5.  What  is  the  cost  of  96  doz.  buttons  at  $1.75  (J  —  i  +  J) 
per  dozen  ? 

0.  Find  the  cost  of  711ff  (J+i)   bushels  oats  at  39c.   per 
bushel. 

7.  Find  the  cost  of  24  boxes  note  paper  at  16f  c.  per  box. 

8.  Find  the  cost  of  24116  Ibs.  bran  at  $1.20  per  cwt. 

9.  Find  the  cost  of  1750  Ibs.  soap  at  5£c.  per  pound. 

10.  Find  the  cost   of   131   Ibs.    coffee  at   16Jc.    per  pound. 


11.  Find  the  cost  of  60£  Ibs.  crackers  at  12Jc.  per  pound. 

12.  Find  the  cost  of  4880  Ibs.  feed  @  75c.  per  cwtf  . 

JT5.  Find  the  cost  of  20  half  -barrels  fish  at  $4.25  per  half- 
barrel.  At  $5.35  (J-f  A)  Per  half  -barrel. 

U.  Find  the  cost  of  75  books  at  25c.  each. 

16.  Find  the  cost  of  36  pairs  shoes  at  $2.25  per  pair.  At  $2.50 
per  pair. 

16.  Find  the  cost  of  3019  Ibs.  bran  at  62|c.  per  cwt.,  and 
24375  Ibs.  feed  at  $1.05  per  cwt. 


PERCENTAGE. 


392.  Percentage  is  a  term  applied  to  all   operations  in 
which  100  is  used  as  the  basis  of  computation. 

It  is  also  the  name  given  to  any  number  of  hundredths  of  a  number. 

393.  Per  Cent.   (%)  is  an  abbreviation  of  the*  Latin  per 
centum,  meaning  on  or  by  the  hundred. 

Thus,  5%  means  5  of  every  hundred,  or  5  hundredths  (T£7,  or  .05). 

394.  Any  per  cent,  may  be  expressed  in  the  form  of  a  decimal 
or  fraction. 

Thus  5  per  cent.  =  5^  =  5  hundredths  =  .05  —  T^  =  •&.     The  first  two 
forms  are  used  in  the  statements  of  questions ;  the  others  in  the  operations. 

395.  In  percentage,  three  elements  are  considered,  viz  :  the 
Base,  the  Rate,  and  the  Percentage.     Any  two  being  given,  the 
other  can  be  found. 

396.  The   Percentage  is  the  result  obtained  by  taking  a 
certain  number  of  hundredths  of  a  number. 

397.  The  Base  is  the  number  of  which  a  certain  number  of 
hundredths  are  taken. 

398.  The  Rate  is  the  number  of  hundredths,  or  the  num- 
ber per  cent. 

Thus,  in  the  statement,  6%  of  300  is  18,  the  Percentage  is  18,  the  Base 
300,  and  6  per  cent.  (.06)  is  the  Rate. 

399.  Applications  of  Percentage. — The  principles  of  per- 
centage are  applied  to  many  of  the  most  common  business  trans- 
actions.   Among  the  most  important  of  these  are  Trade  Discounts, 
Commission,  Insurance,  Profit  and   Loss,  Duties,  Interest,  and 
Exchange. 


142  PERCENTAGE.  [Art.  40O. 

400.  To  find  the  percentage,  the  base  and  rate  being 
given. 

Ex.  What  is  6  per  cent,  of  300  ? 

OPERATION.  ANALYSIS.— $%  means  6  himdredths.    6%  of  300  is 

300  Base.  equivalent  to  .06  of  (or  times)  300.     .06  x  300  =  18.    The 

.06  Rate.  percentage  is  the. product  of  two  factors,  the  base  and 

-£  , ,  the  rate. 

Or,  \%  of  300  is  3,  and  6$  is  6  times  3,  or  18. 
To  find  1  %  of  any  number,  place  the  point  two  places  to  the  left. 

401.  EULE. — 1.    To  find  the  percentage,  multiply   the 
base  by  the  rate  expressed  decimalli/. 

EXAM  PLES. 

402.  What  is  What  is  What  is 

1.  .03x1728?  6.  8%  of  $414?  11.   16$  of  $375.60  ? 

2.  .16  times  375  ?  7.  .08  of  8716  ?  12.  8%  of  $414.60  ? 

3.  4%  of  448  ?  8.  1.12  x  $575  ?  18.  6%  of  $875.75  ? 

4.  6  per  cent,  of  387?  9.  107%  of  $385  ?  14.  113%  of  $913.25  ? 

5.  .06  of  945  ?  10.  9%  of  $456  ?  15.  32%  of  $485.50  ? 

16.  What  is  the  difference  between  2$%  of  $16000  and  5%  of 
$8475  ? 

17.  A  merchant  bought  goods  amounting  to  $375.60,  and  sold 
them  so  as  to  gain  30%  of  the  cost ;  how  much  did  he  gain  ? 

18.  A  lawyer  collected  $2875,  and  charged  5%  for  his  services ; 
how  much  did  he  retain  for  his  services,  and  how  much  did  he 
pay  over  ?    The  amount  paid  over  is  what  per  cent,  of  the  amount 
collected  ?  * 

19.  An  agent  sells  a  house  and  lot  for  $16450,  and  receives  2% 
for  his  services  ;  what  does  he  pay  to  the  owner  of  the  property  ? 

20.  What  is  the  duty,  at  25%  of  the  value,  on  twelve  watches 
worth  $75  each  ? 

21.  Jan.  10,  a  merchant  buys  a  bill  of  goods  amounting  to 
$876.40  on  the  following  terms  :  4  months,  or  less  6%  if  paid  in 
10  days.     How  much  would  settle  the  bill  Jan.  18  ? 

22.  A  merchant,  failing  in  business,  pays  43%  of  his  indebted- 
ness.    He  owes  A  $3750,  and  B  $6280.     How  much  does  he  pay 
each  ? 

*  In  order  to  prepare  the  student  for  examples  in  which  the  conditions  are  the  reverse  of 
those  in  this  example,  the  teacher  should  ask  oral  questions  similar  to  the  above  in  all 
examples  in  which  an  amount  or  difference  is  involved. 


Art.  402.]  PERCENTAGE.  143 

23.  A  commission  merchant  sold  450  barrels  of  flour  at  $5.30 
per  barrel.     How  much  should  he  send  to  the  miller,,  if  he  charges 
2£  per  cent,  for  making  the  sale  ? 

24.  A  manufacturer's  list  price  of  cans  is  $2.50  per  dozen.     He 
sells  a  dealer  48  dozen  at  a  discount  of  30%.     How  much  does  he 
receive  for  them  ? 

25.  A  broker  sells  merchandise  amounting  to  $916.64,  at  a 
commission  of  1-J  per  cent.     What  is  his  commission  ? 

OPERATION. 

ANALYSIS.— \%  of  $916.64  is  $9.166,  to  which  add  \ 
1.146  \%.  of  itself  as  in  the  operation.     (See  Art.  389.) 

10.312  1£#. 
According  to  the  above  method,  find 

26.  1%  of  $375.60.  29.  If  (£  +  £)%  of  $287.96. 

®7-  i  (i  +  i)#  of  $875-  so-  l?%  of  *5275. 

28.  1|%  of  $1176.40.  81.  If  %  of  $3075.75. 

32.  A  contracts  to  make  an  excavation  containing  3456  cubic 
yards,  at  28^  per  cubic  yard  for  earth,  and  $1.20  per  cubic  yard 
for  rock.  When  completed,  it  is  found  that  16%  is  rock,  and  the 
remainder  earth.  How  much  does  he  receive  for  the  work  ? 

4O3.  When  the  rate  is  an  aliquot  part  of  100,  it  is  generally 
more  convenient  to  use  the  equivalent  fraction.  Thus, 

=  .16f=f  6J:%  =  .061  =  TV. 

=  .12i=  i.  5%    =.06    =  ^. 

25%    =.25    =  j,          10%    =.10    = 
=  .20    = 


EXAMPLES. 

4O4.  AVhat  is  What  is  What  is 

1.  J  of  1728- ?  5.  331%  of  $375  ?        9.  50%  of  $487.20  ? 

2.  25%  of  3472  ?       0.   12J%  of  $848  ?      70.  5%  of  $9742  ? 

#.  .25  of  6418?        7.  2 £%  of  $6480?      77.   10%  of  $1764.30  ? 
4.  T^t  of  7264  ?       #.  20%  of  $9875  ?      12.  37|%  ({)  of  $875.60  ? 
13.  From  a  bill  of  goods  amounting  to  $475.60,  5%  is  deducted 
for  cash.     What  is  the  net  amount  of  the  bill  ? 

OPERATION. 

475 
23 


451 


ANALYSIS.—  5$  is  ^.     To  divide  by  20,  divide  by  2  and 
place  the  quotient  figures  one  place  to  the  right. 


82 


144  PERCENTAGE.  [Art.  404. 

14.  A  commission  merchant  pays  $2375.40  for  a  quantity  of 
grain,  and  charges  %\%  for  his  services.     What  is  the  total  cost  ? 

15.  Mr.  B's  tax  is  $175.60.      If  the  collector  is  allowed  5fe 
additional,  what  is  the  total  amount  paid  ? 

16.  The  gross  amount  of  a  bill  of  tinware  is  $97.40.     What  is 
the  net  amount,  if  the  trade  discount  is  ?&\%  ? 

17.  A  house  is  sold  for  $16400,  and  25%  of  the  purchase  money 
is  allowed  to  remain  on  bond  and  mortgage.    What  is  the  amount 
of  the  mortgage  ? 

18.  A  house  worth  $7200  is  insured  for  62|-%  (f )  of  its  value. 
What  is  the  amount  of  the  insurance  ?     What  is  its  cost  at  \%  ? 

405.  To  find  the  base,  the  percentage  and  rate  being 
given. 

Ex.    18  is  6^  of  what  number. 

OPERATION.  ANALYSIS.— The  question  "18  is  6%  of  what  num- 

Kate.  Percentage,      fcer?"  is  equivalent  to  "  18  =  .06  x  what  number?"     If 
£____          18  is  the  product  of  two  factors  and  one  of  the  factors  is 
Base     300         *06>  the  other  may  be  found  by  dividing  18  by  .06. 

Or,  since  the  Percentage  =  the  Base  x  the  Rate,  the 
Base  =  the  Percentage  -5-  the  Rate. 

Or,  if  18  is  Q%  of  a  certain  number,  \%  is  \  of  18,  or  3  ;  and  the  number, 
or  100^,  is  100  times  3,  or  300. 

NOTE. — The  student  should  remember  that  the  Percentage  =  the  Base  x 
the  Rate,  and  that  when  the  Percentage  and  one  of  its  factors  are  given,  in  the 
operation  of  finding  the  other,  the  Percentage  becomes  the  dividend  and  the 
given  factor,  the  divisor.  If  the  relation  between  the  given  terms  is  indicated 
in  the  form  of  an  equation,  the  student  will  have  no  difficulty  in  determining 
which  number  should  be  the  divisor  in  solving  the  great  variety  of  examples 
which  occur  in  percentage  and  its  applications. 

406.  RULE. — To  find  the  base,  divide  the  percentage  by 
the  rate  expressed  decimally. 

EXAMPLES. 

407.  Find  the  unknown  term  in  the  following  equations: 

1.  48  —  6  times .  6.  §%  of  380  = . 

2.  48  =  .06  x .  7.  72  is  4^  of . 

8.  324  is  Q%  of .  8.  184  =  .23  x . 

4.  448  =  .08  of .  9.  175  is  .07  times , 

5.  .04  of =  375.  10.  $576  x  .06  = . 


Art.   4O7.] 


PERCENTAGE.  145 


11.  $324  =  8$  of .  lo.  $144  =  |  of . 

jf;2.  $144  —  .16  x .  16.  12J$  of  $475  = . 

13.  52$  of  $440  = .  17.  $325  is  13$  of . 

1£.  $875  =  25$  of .  18.  $48.60  is  2^$  of . 

19.  The  product  of  two  factors  is  75  ;  if  one  of  the  factors  is 
.03,  what  is  the  other  factor  ? 

20.  The  percentage  is  60,  and  the  rate  2 <J$  ;  what  is  the  base  ? 

NOTE. — Use  the  fractional  method  when  convenient.         See  Art.  4O2. 

21.  $18.08  are  5$  of  what  ?  25.   165/Y.  are  33J$  of  what  ? 
£#.  $324  are  10$  of  what  ?  #0.   £240  are  3%%  of  what  ? 
23.  $37.56  are  2£$  of  what  ?  #7.  $12.25  are  6J$  of  what? 
&£.  $17.28  are  25$  of  what  ?  28.  $96  are  |%  of  what  ? 

#0.  Mr.  A  invests  42$  of  his  capital  in  real  estate,  and  has 
$53070  left ;  what  is  his  capital  ? 

ANALYSIS. — 100$  of  any  amount  is  the  amount  itself.  If  42$  of  his  capi- 
tal is  invested  in  real  estate,  the  remainder,  $53070,  must  he  58$  (100$  —42$) 
of  his  capital. 

30.  A  has  35$  of  his  property  invested  in  stocks,  10$  in 
horses  and  cattle,  18$  in  grain,  and  the  remainder,  which  is 
$24235,  in  real  estate.  What  is  the  total  value  of  his  property  ? 

,  31.  A  horse  was  sold  for  $658,  which  was  16|  $  more  than  it 
cost ;  what  was  the  cost  ? 

NOTE.— The  cost  of  the  horse  was  |gg,  or  100$  of  itself ;  since  the  gain 
was  16f  $  of  the  cost,  the  selling  price  (the  cost  plus  the  gain)  was  116f  $  of 
the  cost.  $658  is  116f  $  of  what  number  ? 

What  number  increased  by  What  number  decreased  by 

32.  25^  of  itself  is  500  ?  35.  5$  of  itself  is  $307.80  ? 

83.  8$  of  itself  is  $1004.40  ?  36.  40$  of  itself  is  3726  ? 

34.  125$  of  itself  is  999  ?  37.  25$  of  itself  is  $342,60  ? 

38.  A  merchant  sells  goods  for  $555.50,  which  is  10$  more 
than  they  cost  him.     What  did  they  cost  ? 

39.  In  a  cargo  of  oranges,  consisting  of  4275  boxes,  33^$  are 
damaged.     How  many  are  damaged  ? 

40.  A  bankrupt   whose   assets  are   $23625,  pays   40$   of  his 
liabilities.     What  are  his  liabilities  ? 

41.  A  farm  was  sold  at  a  commission  of  1J$.     If  the  agent's 
commission  was  $80.80,  what  was  the  price  of  the  farm  ? 


146  PERCENTAGE.  [Art.  4O8. 

408.  To    find  the  rate,  the  percentage  and  base  being 
given. 

Ex.    18  is  what  per  cent,  of  300  ? 

OPERATION.  ANALYSIS. — The   question    "18   is  what .  per 

Base.  Percentage.  Rate.        cent>  of  300  »  is  equivalent  to  "  18  =  how   many 

300  )  18.00  (  .06         hundredths  times  300."     If  18  is  the  product  of 

two  factors,  and  one  of  the  factors  is  300,  the 

other  may  be  found  by  dividing  18  by  300.     To  find  the  rate  per  cent.,  the 
quotient  must  be  produced  in  hundredths. 

Or,  since  the  Percentage  =  the  Base  x  the  Rate,  the  Rate  =  the  Percentage 
-T-  the  Base,     18  -s-  300  =  .06  (6$),  the  required  per  cent. 
Or,  18  is  dfr  or  ^  of  300.     /0  =  Tfo,  or  6  % . 

409.  RULE. — To  find  the  rate,  divide  the  percentage  by 
the  base. 

NOTE. — In  finding  the  rate,  to  produce  a  quotient  of  hundredths,  make 
the  decimal  places  of  the  dividend  exceed  those  of  the  divisor  by  2. 

EXAMPLE  S. 

410.  -1.  The  product  of  two  numbers  is  375  ;  if  one  of  the 
numbers   is   30000,  what   is   the   other   number  ?      Express   the 
answer  in  hundredths. 

Find  the  unknown  term  in  the  following  equations  : 

2.  75  is  what  per  cent,  of  375?        6.  $12.50  is  what  %  of  $1000? 

3.  144  =  .**  times  1728.  7.  $232.50  is  what  %  of  $3720? 

4.  72  =  .**  x  3456.  8.  $60.40  is  what  %  of  $2416? 

5.  165  ft.  is  what  %  of  5280  ft.  ?      9.  $21.20  is  what  %  of  $1484? 

10.  The  assets  of  a  bankrupt  are  $27387,  and  his  liabilities 
$82161 ;  what  %  of  his  indebtedness  can  he  pay  ? 

\11.  A  merchant  paid  for  goods  $345  and  sold  them  for  $258.75; 
the  loss  is  what  %  of  the  cost  ? 

12.  If  a  paymaster  receives  $150000  from  the  treasury,  and 
fails  to  account  for  $225  thereof,  what  is  the  percentage  of  loss  to 
the  government  ? 

IS.  If  the  rate  is  20^  and  the  percentage  440,  what  is  the  base? 

14.  $640  being  increased  by  a  certain  %  of  itself  equals  $720 ; 
required  the  rate  %. 

15.  A  person  owing  me  $2092,  pays  only  $1150.60.     What  % 
do  I  lose  on  the  debt  ? 


•    . 

Art,41O.]  PERCENTAGE.  147 

16.  A   house,    insured  for    $16500   in  several   companies,    is 
damaged  by   fire  to  the  extent  of  $7260.     What  fc  of  its  insur- 
ance will  each  company  pay  ? 

17.  A  tax  of  $18480  is  levied  upon  a  township  whose  valuation 
is  $3,696,000.     What  is  the  rate  of  the  tax,  and  what  tax  should 
a  man  pay  whose  assessment  is  $8500  ? 

•^  18.  The  annual  interest  on  a  mortgage  of  $7500  is  $337.50. 
What  is  the  rate  per  cent.  ? 

19.  A  merchant  with  a  capital  of  $24000  gains  $3840  in  one 
year.     His  gain  is  what  per  cent,  of  his  capital  ? 

20.  A  bankrupt  has  liabilities  to  the  amount  of  $12600  and 
his  assets  are  only  $7087.50.     What  %  dividend  will  he  pay  ? 

21.  The  dividend  of  a  manufacturing  company,  whose  capital 
stock  is  $125000,  is  $6000.     What  %  does  it  pay  ? 

22.  A  man's  salary  is  $1600  per  year  and  his  living  expenses 
$1300.     What  %  of  his  salary  does  he  save  ? 

23.  A  house  cost  $8000,  and  rents  for  $750  per  year.     If  the 
taxes  and  other  expenses  are  $230  per  year,  what  %  does  it  pay  on 
the  investment  ? 

24.  The  cost  of  insuring  a  cargo  for  $8500  is -$63. 75.     What 
is  the  rate  of  the  insurance  ? 

25.  I  meter  =  1.0936  yards.     The  meter  is  what  %  greater 
than  the  yard  ? 

26.  A  man  subscribes  for  36  shares  ($100  each)  of  a  gas  com- 
pany.    He  pays  in  $1980.     What  %  is  still  due  ? 

27.  The  Avoirdupois  pound  (334)   is  what  %  greater  than 
the  Troy  pound  (336)  ? 

28.  A  merchant  bought  a  quantity  of  goods  for  $425.     Being 
damaged  he  sold  them  for  $340.     What  %  of  the  cost  did  he  lose  ? 

29.  A  horse  and  wagon  are  worth  $600.     What  is  the  value  of 
each,  if  the  wagon  is  worth  87^%  as  much  as  the  horse  ? 

SO.  A  man  bought  a  watch  for  $160  and  sold  it  for  $180.    His 
gain  was  what  %  of  the  cost  ? 

31.  A  tea  merchant  mixes  40  Ibs.  tea  at  45c.  per  pound  with 
50  Ibs.  at  27c.  per  pound.     He  sells  the  mixture  at  42c.  per  pound. 
What  %  profit  on  the  cost  does  he  make  ? 

32.  A  consignment  of  flour  was  sold  for  $3148,  of  which  $3124. 39    ^ 
were  the  net  proceeds.     What  was  the  rate  %  of  the  commission  ? 

33.  Mr.  A's  house  is  worth  $12500.     He  pays  $30  for  insuring 
it  for  |  of  its  value.     What  per  cent,  does  he  pay  ? 


148  PERCENTAGE.  [Art.  411 


REVIEW    EXAMPLES. 

411.     1.  What  is  116$  of  1200  ? 

2.  144  is  120%  of  what  number  ? 

3.  375  is  what  %  of  300  ? 

4.  Find  95$  of  $1260. 

5.  Of  what  number  is  275,  100$  ? 

6.  $187.50is2i$  of  what  ? 

7.  What  will  be  the  charge  for  insuring  a  house    for  $4500 
atf$? 

8.  A  merchant  buys  one  gross  jars  for  $36.     At  what  price 
must  he  sell  them  apiece  to  gain  20$  on  the  cost  ? 

9.  The  assets  of  a  bankrupt  are  $67850,  which  sum  is  43$  of 
his  debts.     What  are  his  debts  ? 

10.  A  tax  collector,  whose  average  commission  was  3J$,  received 
$892.08  for  his  services.     How  much  did  he  collect  ? 

11.  At  what  price  must  an  article,  which  cost  $4.80,  be  sold  so 
as  to  gain  16$  of  the  cost  ? 

12.  A  clerk-  spends  48$  of  his  income,  and  saves  $598.     What 
is  his  income  ? 

13.  A's  property  is  assessed  at  $7500,  and  the  rate  of  taxation 
is   $2.165   on   $100.     What   is   his   tax,  including  a  commission 
of  1$? 

14.  At  1|$,  the  premium  for  insuring  a  factory  was  $178.20. 
Find  the  amount  of  the  insurance. 

15.  A  bank  collected  a  draft  of  $9375.16.     What  were  the  pro- 
ceeds, the  charge  for  collection  being  \%  ? 

16.  A  consignment  of  cheese  was  sold  for  $375.60,  of  which 
$365.27  were  the  net  proceeds.     What  was  the  rate  of  the  com- 
mission ? 

17.  A  commission  merchant  sold  24160  pounds  of   leather  at 
29  J  cents  a  pound,  paid  transportation  $60.40,  cartage  $20,  his 
commission  being  2|-$,  and  his  charge  for  inspection  $20.     What 
were  the  net  proceeds  ? 

18.  A  bankrupt  who  is  paying  36$  of  his  debts,  divides  among 
his  creditors  $44442.     What  do  his  debts  amount  to,  and  how 
much  does  he  pay  a  creditor  whom  he  owes  $3648  ? 

19.  A  merchant  buys  a  bill  of  dry  goods,  Apr.  16,  amounting 
to  $6377.84,   on  the  following  terms  :  4  months,  or  less  5$  30 


Art.  411.]  PERCENTAGE.  149 

days.  How  much  would  settle  the  account  May  16  ?  The  amount 
paid  May  16,  is  what  per  cent,  of  the  full  amount  of  the  bill  ? 
The  above  discount  is  equivalent  to  what  rate  per  cent,  per  an- 
num ? 

20.  Mar.   16,  a  merchant  buys  a  bill  of  goods  amounting  to 
$2475  on  the  following  terms  :  4  months,  or  less  5%  if  paid   in 
30  days.     Apr.  15,  he  makes  a  payment  of  $1000,  with  the  under- 
standing that  he  is  to  have  the  benefit  of  the  discount   of   5%. 
With  what  amount  should  he  be  credited  on  the  books  of  the 
seller  ?    How  much  would  be  due  July  16,  the  expiration  of  the  4 
months  ? 

NOTE. — As  in  Ex.  19,  the  amount  paid  within  30  days  is  95$  of  that  part 
of  the  bill  which  it  settles  or  cancels. 

21.  A  bought  a  bill  of  merchandise  July  24,  1879,  amounting 
to  $6287.45  on  the  following  terms  :  6  months,  or  less  4%  30  days. 
He  paid  on  account  Aug.  23,  1879,  $5000,  with  the  understand- 
ing that  the  payment  would  cancel  an  equitable  amount  of  the 
bill.     How  much  was  due  Jan.  24,  1880  ? 

22.  Paid  for  transportation  $664.95  on  an  invoice  of  goods 
amounting  to  $8866.     What  per  cent,  was  the  value  of  the  goods 
thereby  increased  ?    What  per  cent,  must  be  added  to  the  invoice 
cost  to  make  a  profit  of  20%  on  the  full  cost  ? 

28.  What  is  3%  of  £247  13s.  6d.  ? 

OPERATION.  ANALYSIS. — Multiply  the   number  of  each  de- 

s.        d.  nomination  by  .03,  as  in  the  margin,  and  then  re- 

247       13         6  duce  the  decimal  parts  to  integers  of  lower  denom- 

.03  inations  (289). 

-P  71  4.1        3Q To  ^r'  re(luce  shillings  and  pence  to  the  decimal  of 

1     £      '  a  pound  (see  note,  Ex.  12,  Art.  342),  take  the  re- 

quired per  cent.,  and  reduce  the  decimal  result  to 
S.    8|.59  lower  denominations.     Thus, 

12  £247  13s.  Qd.  =  £247.675 

d  ~Y~26  £247.675  x  .03  =  £7.43025  =  £7  8s.  Id. 

When  the  rate  per  cent,  is  an  aliquot  part  of 

100,  use  the  equivalent  fraction  (4O3).  Thus,  5%  of  £247  13s.  6d.  =  -fa  of 
£247  13s.  Qd.  =  £12  7s.  Sd. 

24.  Find  3%  of  £384.  28.  Find  4^  of  £75  12s.  Qd. 

25.  Find  S%  of  £440  16s.  29.  Find  10%  of  £37  8s.  9rf. 

26.  Find  $%  of  £375.  80.  16s.  is  %\%  of  what  ? 

27.  Find  2       of  £64  16s.  81.  £1  8s.  4d.  is  4     of  what  ? 


150  PERCENTAGE.  [Art.  412. 


PROFIT     AND     LOSS. 

412.  Profit  and  Loss  treats  of  the  gains  (profits)  and  losses 
which  arise  in  business  transactions. 

The  profit  or  loss  is  always  estimated  on  the  cost  price,  or  the  amount 
invested.  Discounts  are  reckoned  on  the  marked  or  asking  price.  (See 
Art.  415.) 

413.  The  difference  between  the  cost  of  goods  and  the  price 
at  which  they  are  sold  is  a  profit  or  a  loss, — profit  if  the  selling 
price  is  the  greater,  loss  if  the  cost  is  the  greater. 

EXAMPLES. 

414.  1.  A  man  purchased  a  horse  for  $250,  and  sold  it  at  a 
gain  of  16$.     What  was  the  gain  ?     (Gain  =  .16  x  cost.) 

2.  A  merchant  sold  goods  that  cost  1325  at  an  advance  of  12$; 
what  was  the  selling  price  ?      (Gain  =  .12  x  cost,   and  selling 
price  =  cost  -f  gain  ;  or,  selling  price  =  1.12  x  cost.) 

3.  Bought  a  farm  for  $3600,  and  sold  it  at  an  advance  of  25$; 
what  was  the  gain  ? 

NOTE. — If,  as  in  the  above  example,  the  rate  per  cent,  is  an  aliquot  part 
of  100,  it  is  more  convenient  to  use  the  equivalent  fraction  (4O3).  Thus, 
25%  =  .25  =  £  ;  gain  —  \  of  cost. 

4>  Cloth  is  bought  at  $6  per  yard,  and  sold  at  a  loss  of  20%. 
What  is  the  selling  price  ?  (Selling  price  =  $•  of  cost.) 

5.  Bought  a  house  for  $3475  ;  at  what  price  must  it  be  sold  to 
gain  36$  ? 

6.  Purchased  flour  at  $6.25  per  barrel ;  at  what  price  must  it 
be  sold  to  gain  20$  ? 

7.  If  I  buy  hats  at  $27^  per  dozen,  at  what  price  must  I  sell 
them  apiece  to  gain  33^$  ? 

8.  A  factory  which  cost  $8775   was  sold   at  a  gain  of  16$. 
What  was  received  for  it  ? 

9.  If  silk  costs  $1.68  per  yard,  and  is  sold  at  an  advance  of 
12-|%  what  is  the  profit  per  yard  ? 

10.  A  merchant  purchased  goods  to  the  amount  of  $8735,  and 
sold  them  at  a  loss  of  12$  ;  what  was  his  loss  ? 

11.  Bought  125  barrels  of  flour  for  $600.    If  sold  at  an  advance 
of  15$,  what  was  the  profit  per  barrel  ? 


Art.  414.]  PROFIT    AND     LOSS.  151 

12.  A  lot  of  dry  goods  was  sold  at  an  advance  of  18%.    If  the 
gain  was  $436.50,  what  was  the  cost  ?    (Gain  =  .18  x  cost ;  hence, 
gain  -f-  .18  =  cost.) 

13.  A  farm  was  bought  for  $7200,  and  sold  at  a  gain  of  $900  • 
what  was  the  gain  per  cent.  ?     (Gain  =  gain  %  x  cost ;    hence, 
gain  %  =  gain  -j-  cost.) 

14.  A  man  paid  for  merchandise  $875,  and  sold  it  for  $1015 : 
what  per  cent,  did  he  gain  ? 

15.  A  man  paid  for  merchandise  $1015,  and  sold  it  for  $875  ; 
what  per  cent,  did  he  lose  ?. 

16.  Find  the  rate  %  of  profit  on  goods  bought  for  $324  and  sold 
for  $364.50. 

17.  A  painting  was  sold  for  $2343,  at  a  gain  of  32%  ;  what  was 
the  cost?     [Selling  price  =  1.32  (100%  +  32%)  x  cost;   hence, 
cost  =  selling  price  -?-  1.32.] 

18.  Find  the  cost  of  goods  sold  at  an  advance  of  12-J-%,  being 
a  profit  of  $76. 

19.  How  much  was  paid  for  a  farm  sold  for  $9878,  at  12% 
below  cost  ? 

20.  What  is  the  profit  on  iron  cold  for  $4520,  at  an  advance 
of  13%  on  cost  ? 

21.  What  is  the  selling  price  of  tea  which  cost  32  cents  per 
pound  and  is  sold  at  a  profit  of  37£%  ? 

22.  Sold  drugs  for  $168,  at  an  advance  of  75%  ;  what  was  the 
profit  ? 

23.  A  merchant  sold  for  $2576  a  lot  of  dry  goods  for  which 
he  paid  $3360.     What  was  the  per  cent,  of  loss  ? 

24.  A  mixture  is  made  of  1  gallon  of  wine  at  50  cents  a  gallon, 
3  at  90  cents,  4  at  $1.20,  and  12  at  40  cents.     What  per  cent, 
would  be  gained  by  selling  the  mixture  at  $1.60  a  gallon  ? 

25.  If,  by  selling  tea  at  47 J-  cents  per  pound,  1  lose  5%,  at 
what  price  must  I  sell  it  to  gain  15%  ? 

26.  If,  by  selling  goods  for  $126,  I  lose  16%,  what  per  cent, 
would  I  have  lost  or  gained  if  I  had  sold  them  for  $168? 

27.  A  merchant's  price  is  25%  above  cost  price.     If  he  allows  a 
customer  a  discount  of  12%  on  his  bill,  what  per  cent,  profit  does 
he  make  ? 

28.  If  cloth,  when   sold   at   a   loss    of   25%,    brings   $5   per 
yard,    what   would   be   the   gain   or  loss    per   cent,    if    sold    at 
$6.40  per  yard? 


152  PERCENTAGE.  [Art.  414. 

29.  Goods  that  cost  $168  are  sold  at  an  advance  of  25%  ;  what 
is  the  selling  price  ? 

SO.  At  what  price  must  ribbon  be  sold  per  yard  so  as  to  gain 
20%,  if  22£  yards  cost  $6.75  ? 

31.  A  merchant  gave  $25000  for  seven  houses.  What  per 
cent,  does  he  gain  by  selling  them  at  $7000  each  ? 

82.  A  woman  buys  a  certain  number  of  apples  at  the  rate  of  3 
for  1  cent,  and  as  many  more  at  the  rate  of  2  for  1  cent.  What 
per  cent,  does  she  gain  or  lose,  if  she  sells  them  at  the  rate  of  5 
for  2  cents  ? 

38.  Eggs  are  bought  at  27  cents  per  dozen,  and  sold  at  the 
rate  of  8  for  25  cents.  What  is  the  per  cent,  of  profit  ? 

34.  A  merchant  by  selling  goods  for  $364,  loses  9%.     For  what 
ought  they  to  be  sold  to  gain  8%  ? 

35.  A  drover  bought  160  sheep  for  $400,  and  sold  f  of  them  at 
$2.25  each.     At  what  price  must  he  sell  the  remainder  so  as  to 
gain  10%  on  the  whole  ? 

36.  A  merchant  sells  goods  to  a  customer  at  a  profit  of  60%, 
but  the  buyer  becoming  bankrupt  pays  only  70^  on  the  dollar. 
What  %  does  the  merchant  gain  or  lose  by  the  sale  ? 

87.  If  a  merchant  adds  to  the  cost  price  of  his  goods  a  profit  of 
12}%,  what  is  the  cost  of  an  article  which  he  sells  for  $7.20  ? 

88.  Sold  a  horse  at  a  gain  of  33J%,  and  with  the  proceeds  pur- 
chased another  horse,   which  I  sold  for  $120,  at  a  loss  of  20%. 
What  was  the  gain  or  loss  ? 

89.  A  merchant's  retail  price  for  boots  is  $4. 75  per  pair,  by 
which  he  makes  a  profit  of  33J%.     He  sells  to  a  wholesale  customer 
at  a  discount  of  20%  from  the  retail  price.     What  per  cent,  does 
he  gain  or  lose,  and  what  does  he  receive  per  pair  ? 

40.  40  head  of  cattle  weighing  52770  pounds  are  purchased  in 
Chicago  at  $4.80  per  cwt.,  and  are  sold  in  New  York  at  10 \  cents 
per  pound,  to  dress  56  pounds  to  the  hundred- weight.     What  was 
the  total  cost  ?    The  total  selling  price  ?    What  is  the  gain  per 
cent.,  making  no  allowance  for  transportation  ? 

NOTE. — The  quantity  bought  or  sold  does  not  affect  the  gain  or  loss  per 
cent. 

41.  A  speculator  sold  two  building  lots  for  $4800  each.     On 
one  he  gained  20%,  and  on  the  other  he  lost  20%.     Did  he  gain  or 
lose,  and  how  much  ? 


Art.  415.]  DISCOUNTS.  153 


DISCOUNTS. 

415.  It  is  customary  in  many  branches  of  business  for  manu- 
facturers and  dealers  to  have  fixed  price-lists  of  certain  kinds  of 
merchandise  ;  and  when  the  value  changes,  instead  of  changing  a 
long  price-list,  the  rate  of  discount  is  changed.     The  fixed  price 
is  called  the   List  Price,  and   the   discount   allowed   the  Trade 
Discount. 

Books  are  usually  sold  by  publishers  and  jobbers  at  certain  discounts  from 
the  retail  prices. 

416.  Many  kinds  of  merchandise  are  sold  at  " time"  prices, 
subject  to  certain  rates  of  discount  if  paid  at  an  earlier  period. 

1.  Thus,  the  following  or  similar  announcements  are  usually  found  upon 
the  bill-heads  of  wholesale  dealers  :  "  Terms,  4  months,  or  30  days  less  5%"; 
or,  "  Terms  60  days,  or  \%  discount  in  30  days,  or  2%  discount  in  10  days.5' 

2.  In  the  same  business  house,  certain  goods  are  sold  on  long  credit,  and 
others  on  short  credit. 

3.  When  no  rate  of  discount  has  been  offered,  merchants  are  generally 
willing,  when  bills  are  paid  before  maturity,  to  deduct  the  interest  on  the 
amount  of  the  bill  for  the  remainder  of  the  time  at  the  legal  rate  per  annum. 

Ex.     The  list-price  of  a  scale  is  $80  ;  what  is  the  net  price  if 
a  discount  of  25%  and  10%  is  allowed  ? 

OPERATION. 

$80  List-price  ANALYSIS. — The  first  rate  of  discount  is  reckoned 

~  ~  upon,  and  deducted  from  the  list  price,  and  the  others 

'/»>  or  ?.  aie  dg^uc^  from  the  successive  remainders. 

60  The  result  is  not  affected  by  the  order  in  which  the 

6  10%   or  1  .  discounts  are  taken.     A  discount  of  25%  and  10%  is  the 

same  as  a  discount  of  10%  and  25%. 
54  Net-price. 

EXAMPLES. 

417.  1.  The  gross  amount  of  a  bill  of  shoes  is  $82.68.     What 
is  the  net  amount,  the  rate  of  discount  being  5%?     (See  Ex.  1, 
Art.  391.) 

2.  A  stove  is  sold  for  $45  less  30%;  required  the  net  price  ? 
NOTE. — If  the  discount  is  not  required,  multiply  by  .70  (100%  — 30%); 

the  product  will  be  the  net  price.     To  multiply  by  .70,  multiply  by  7  and 
place  the  figures  of  the  product  one  place  to  the  right. 

3.  What  is  the  value  of  466  II.   O.W.  casing  @  45  cts.  per 
pound,  less  1^  per  cent.  ? 


154  PERCENTAGE.  [Art.  417. 

4.  The  gross  amount  of  a  bill  of  mdse.  is  $100.36  ;  what  is 
the  net  amount,  the  rates  of  discount  being  20%  and  10$? 

5.  The  gross  amount  of  a  bill  of  notions  is  $49.  75  ;  what  is  the 
net  amount,  the  rates  of  discount  being  10$  and  10$? 

6.  What  is  the  value  of  12  pair  shoes  @  $1.60  per  pair,  less  5$? 

7.  What  single  discount  is  equivalent  to  a  discount  of  20$ 
and  10$  ? 

ANALYSIS.  —  Represent    the    gross   amount   by   100  % 

OPERATION 

-j  00  (1.00).     20$  (|)  of  100^  =  20$  (.20),  which  subtracted 

'on  on*-i       from  100#   (1'00)'  leaves  8°^    ('80)-     10#   (A>  °f  8°^ 
jfX**     ~~*      (.80)=8$  (.08),  which  subtracted  from  80$  (.80),  leaves 

.80  72$  (.72).     100$  -72  $=28$,  the  direct  discount. 

Qg    1   of  80.  By  the  following  rule,  a  single  discount  can  be  cal- 

culated from  two  discounts  mentally:  From  the  sum  of 
•  **  the  discounts,  subtract  ^  of  their  product.    The  remainder 

.  28  will  be  the  real  discount.     Thus,  20  %  +  10$  =30$  .     30$ 

—2$  (20x10  -^100)  =28$.     When  a  third  discount  is 
given,  combine  it  with  the  result  obtained  from  the  other  two. 

When  the  sums  of  two  or  more  discount  series  are  the  same,  the  series,  in 
which  the  discounts  are  the  most  uniform,  will  produce  the  least  single  dis- 
count; and  the  series,  in  which  the  discount  is  most  concentrated  in  one  dis- 
count, will  produce  the  greatest  single  discount. 

Thus,  a  discount  of  10,  10,  and  10$  is  equivalent  to  27XV$  ;  20,  5,  and  5 
to  27f$  ;  and  25,  2|,  and  2£$  to  28fg$. 


8.  What  single  discount   is   equivalent  to  a  discount  of  15$ 
and  10$?     45$  and  10$?     20$  and  12|$?     60$  and  10$?     75$ 
and  12  J$?    20$,  20$,  and  10$?     60$,  20$,  and  20$? 

9.  The  net  amount  of  a  bill  of  goods  is  $74.20.     What  is  the 
gross  amount,  the  discount  being  30$  ? 

10.  The  net  amount  of  a  bill  of  files  was  $36.75;  what  was 
the  gross  amount,  the  rate  of  discount  being  10$  ? 

11.  A  is  offered  dress  goods  at  26s  cts.  per  yd.,  "4  months  or 
less  6$  cash"  ;  how  many  yards  can  he  purchase  for  $49.82  cash  ? 

The  net  amount  of  a  bill  of  hardware  is  $175.26  ;  what  is 
the  gross  amount,  the  rate  of  discount  being  45$  and  10$  ? 

13.  What  is  the  net  value  of  one  case  prints  containing  2273 
yd.,  @  43  cts.,  less  5$,  cooperage  25  cts.  ? 

14.  A  bill  of  merchandise  amounting  to  $442.38  was  bought 
Aug.  18,  1879,  on  the  following  terms  :  "4  months  or  5$  off  30 
days."     How  much  would  settle  the  bill  Sept.  16,  1879  ? 

15.  What  is  the  net  value  of   a  bill   of   iron   amounting  to 
$1103.75,  at  a  discount  of  45,  10,  and  2  per  cent.? 


Art.  417.]  DISCOUNTS.  155 

16.  What  is  the  net  value  of  1  case  prints  containing  3039 2  yd. 
@  5  cts.  per  yd.,  less  a  discount  of  3%;  cooperage  $.25-? 

17.  What  is  the  difference  on  a  bill  of  $875  between  a  discount 
of  40%  and  a  discount  of  30%  and  10%  ? 

18.  A  bill  of  tinware  is  sold  at  the  following  discounts:  $74.20 
at  20%  and  10% ;  $43.75  at  40%  and  5% ;  $69  at  33£%  and  10% ; 
and  $49.17  net.     What  is  the  total  net  amount  of  the  bill  ? 

19.  A  bill  of  dry  goods  amounting  to  $914.37  is  sold,  Aug.  19, 
on  the  following  terms  :  "  60  days,  or  less  1%  if  paid  in  30  days, 
or  less  2%  if  paid  in  10  days."     How  much  would  settle  the  bill 
Sept.  18  ?    How  much  Aug.  27  ? 

20.  Of  a  bill  of  hardware,  $61.51  are  sold  at  a  discount  of  60 
and  5%;  $18.75  at  a  discount  of  10%;   $16.86  at  a  discount  of 
12 J%;  $44.25  at  a  discount  of  40  and  5%;  $29.60  at  a  discount 
of  40,  12|,  and  10% ;  $28.04  at  a  discount  of  55% ;  $16  at  a  dis- 
count of  65,  10,  and  10%;  $18.70  at  a  discount  of  50%;  $19.75 
at  a  discount  of  20%  ;  $18.50  at  a  discount  of  15%  ;  $307.55  at  a 
discount  of  75  and  12 J% ;  $36.61  at  a  discount  of  60  and  10%; 
and  $218.25  net.     What  is  the  total  net  amount  of  the  bill  ? 

21.  Goods  are  bought  at  a  discount  of  30%  from  a  list  price, 
and  sold  at  the  list  price.     What  is  the  gain  per  cent.  ? 

ANALYSIS. — Assuming  $1  as  the  list  price,  the  cost  is  70c.,  selling  price 
$1,  and  the  gain  30c.     30c.  is  what  %  of  70e.  ? 

22.  Books  are  purchased  at  a  discount  of  25%  from  the  list 
price.     What  is  the  gain  per  cent,  by  selling  at  the  list  price  ? 

28.  What  per  cent,  is  gained  by  selling  pans  at  21  cents  apiece, 
that  cost  $2.56  per  dozen  less  20  and  12 \%  ? 

24.  Plows  are  bought  at  a  discount  of  50%  from  the  list  price. 
What  per  cent,  is  gained  by  selling  at  the  list  price  ? 

25.  A.  merchant  purchases  goods  at  a  discount  of  25%  from  the 
list  price.     What  per  cent,  is  gained  by  selling  at  the  list  price  ? 
What  per  cent,  if  goods  are  purchased  at  a  discount  of   33J%  ? 
35%?    25%  and  5%  ?     20%  and  12J%  ?     15%  and  10%  ? 

26.  A  merchant  buys  goods  at  a  discount  of  40  and  20%  from 
the  list  price,  and  sells  at  a  discount  of  30  and  10%.     What  is  the 
gain  per  cent.  ? 

ANALYSIS. — Assume  $1  as  the  list  price,  find  the  net  cost  and  selling 
prices,  and  then  the  gain  per  cent. 


156  PERCENTAGE.  [Art.  417, 

27.  If  a  merchant  buys  goods  at  a  certain  price  10  and  5  off, 
and  sells  them  at  the  same  price,  5  off,  what  per  cent,  profit  does 
he  make  ? 

28.  What  per  cent,  profit  does  a  merchant  make  who  buys  at  a 
discount  of  20,  10,  and  12-|-$,  and  sells  at  the  list  price  ? 

29.  What  must  be  the  marked  price  of  goods  costing  $32,  that 
I  may  deduct  20$  from  it,  and  still  gain  25$  on  the  cost  ? 

ANALYSIS. — First  find  the  selling  price,  and  then  the  marked  or  list  price. 
If  the  cost  is  $32,  the  gain  will  be  25$  (£)  of  $32,  or  $8;  and  the  selling  price 
will  be  $32  +  $8,  or  $40.  If  the  goods  are  sold  at  a  discount  of  20$,  the  sell- 
ing or  net  price  is  80$  of  the  marked  price.  $40  =  .80  of  $50. 

80.  What  must  be  the  asking  price  for  books  that  cost  $1.60, 
in  order  to  abate  20$,  and  still  make  a  profit  of  25$  ? 

31.  What  must  be  the  list  price  of  goods  that  cost  $18,  in 
order  to  make  a  profit  of  33-J-$,  if  they  are  sold  at  a  discount  of 


?.  Find  the  list  price  of  goods  that  cost  $75  and  are  sold  at  a 
Discount  of  60  and  10$,  at  a  profit  of  20$. 

33.  A  manufacturer  sells  his  goods  at  a  discount  of  30  and 
and  thereby  gains  12£$.  What  is  the  list  price,  if  the  cost 
is  $28  ? 

84.  A  hardware  dealer  sells  certain  goods  at  a  discount  of  75 
and  12J$,  and  gains  20$.  What  is  the  list  price,  if  the  cost  is 
$2.80  ? 

35.  What  per  cent,  must  be  added  to  cost  price  in  order  to 
give  a  discount  of  25$,  and  make  a  profit  of  20$  ? 

ANALYSIS. — Assuming  100$  as  the  cost  price,  the  selling  price  is  120$ 
of  the  cost.  120$  is  75$  (100^-25$),  or  f,  of  160$.  160$— 100$  =  60^, 
the  per  cent,  to  be  added  to  the  cost  price. 

2, vi  N^^S.  What  advance  on  cost  would  be  necessary  in  order  to  give 
a  discount  of  20$,  and  still  make  a  profit  of  20$  ? 

87.  At  what  per  cent,  above  cost  must  goods  be  marked,  so 
that  when  sold  at  a  discount  of  5$,  there  would  be  a  profit 
of  25$  ? 

38.  If  goods  are  bought  at  a  discount  of  2  10's  and  a  5  from  a 
manufacturer's  list  price,  and  sold  at  a  discount  of  12£$  (^),  what 
is  the  gain  per  cent.  ? 

I  purchase  books  at  $2  each  less  33^$,  and  5$  for  cash. 
What  is  the  net  cost,  and  what  per  cent,  discount  may  be  given 
on  the  list  price  to  produce  a  net  profit  of  10$  ? 


Art.  418.] 


BILLS. 


15? 


BILLS.* 

418.  A  Bill  is  a  detailed  statement  of  merchandise  sold,  or 
of  services  rendered.     Bills  of  merchandise  state  the  place  and 
date  of  the  sale,  the  names  of  the  buyer  and  seller,  the  terms  of 
the  sale,  the  quantity,  price,  and  distinguishing  marks  and  num- 
bers of  the  merchandise,  and  other  details. 

The  terms  Bill  and  Invoice  are  used  by  many  interchangeably.  The  term 
Invoice  is  applied  more  particularly  to  statements  rendered  by  consignees  to 
commission  merchants,  showing  marks,  numbers,  values,  and  accrued  charges 
of  goods  shipped;  to  bills  rendered  to  jobbers;  and  to  bills  received  from  for- 
eign countries. 

EXAM  PLES. 

419.  Copy  and  extend  the  following  bills  : 


Messrs.  WM.  DOLTON  &  Co., 


(1.  Canned  Goods.) 

WILMINGTON,  DEL.,  Nov.  16,  1889. 

Bought  of  JAMES  MORROW  &  SON. 


Cases. 

2 

1 
1 
2 
1 
1 

Messr 
Int 

Doz. 

3    Ib.  Peaches    -  *    -    -    -    - 
Saco  Corn     

321* 

180 

400 

400 

BUFFAL 
3CHOELI 

allow  no 

9 
* 
* 
* 

* 

00 

** 
** 

** 

** 
50 

$** 

** 

4 
2 
2 
4 

2 
2 

s.  DA 
erest  ct 

3    '     Tomatoes,  B.  &  L.  -    -    - 
2£  '     Col.  Pears     
24  '     Apricots  ------ 
Ctg.     - 
B  Price  per  dozen. 

(2.  Flour.) 

KIEL  GROUSE  &  SONS, 
Bought  of  i 
arged  on  all  accounts  after  30  days.    We 

o,  N.  Y.,  Dec.  6,  1888. 

KOPP  &  MATTHEWS. 
Expressage  or  Exchange. 

20 
25 
25 
25 
15 
5 

Bbls.  Flour  "  Sunlight  "  Sacks  - 
Bbls.    - 
"Victor"      Sacks  - 
Bbls.    - 
"Dakota"    Sacks  - 
'                 "Superior"  Sacks  - 
2177  Ih    S   Mpfll 

$7.05 
7.25 
6.05 
6.25 
5.30 
8.55 
1.20* 
.56b 

Be.  per  bt 

*** 
*** 
*** 
*** 
** 
** 
** 
*** 

** 
** 
** 
** 
*# 
** 

*** 

** 

20  bags 
70     " 

264&  bu.  Oats    
•  $1  .90  per  hundredweight.    b  5 

she!. 

*  It  is  suggested  that  a  part  of  these  bills  bo  reserved  for  review.  One  or  two  of  them  may 
be  given  each  week  as  a  general  exercise. 


158 


PERCENTAGE. 


[Art,  419. 


(3.  "Window  Glass.) 

PITTSBURGH,  May  14,  18S8. 
EUREKA  GLASS  Co., 

Bought  of  CUNNINGHAMS  &  Co. 
Terms  SO  days.— If  not  promptly  paid,  interest  will  be  charged  from  date  of  bill. 


2 

Bx's7x9    "A"      7^ 

*# 

1  ^ 

8v  10                                                                               750 

*** 

** 

2 

9x12               7^ 

** 

1 

10x14               7^ 

* 

** 

19  v  9ft                                                                       850 

** 

2 

13x36               10^ 

** 

** 

Uv  90                                                  fts  o 

** 

1  ^  v  SO                                                                               Q7A 

** 

** 

*** 

Less  60  and  20^     -    -    - 

*** 

** 

(4.  Provisions.) 

CLEVELAND,  0.,  Oct.  9,  18S6. 
Messrs.  L.  C.  MAGAW  &  SON, 

Bought  of  J.  P.  ROBISON  &  Co. 
Terms  Net  Cash.—  No  goods  sold  on  30  days. 


10 

"Rhl«    CJ    M    Pnrlr                                                  1  7°£ 

*** 

"     Mess  Beef                 -                  1076 

** 

** 

5 
3 
1 
1 

"     Hams         90M376b-98«  ****<»    14^ 
"     Shoulders  58     744  -57      ***       9^ 
"     Dr.  Beef     33     241  -22      ***     14^ 
Tc.  Lard                     406  -60      ***     ll/- 

*** 
** 
** 
** 

** 
** 
** 

** 

*** 

** 

Number  of  pieces.    b  Gross  weight.    c  Tare,  or  weight  of  barrel  or  tierce.    d  Net  weight. 


(5.  Fish.) 

GLOUCESTER,  MASS.,  Sept.  28,  1886. 
Messrs.  DANIEL  WEIDMAN  &  Co., 

Bought  of  CLARK  &  SOMES. 
Subject  to  sight  draft  without  notice  after  thirty  days. 


Htl     UPW   ftpn    Prvl                                                 ^  7^ 

** 

~ 

1 

10 
10 
2 
10 
5 
3 

Bbl.  Ex.  *  1  Mackerel      20.00 
Kits  15  Ibs.  Ex.  *1  Mackerel  -    -    -    1.80 
"     20  Ibs.  Bay*  1        "         -    -    -    1.80 
Bbls.«2Shore                "        lg.  -    -  12.00 
Kits  20  Ibs.  ft  2  Shore     "        "    -    -    1.50 
Halfs  New  Labrador  Herring  -    -    -    3.82 
"      Round  Shore          "       -    -    -    2.95 
Box  '88,  ctg.  in  Boston  <8° 

#* 

## 
#* 
* 

«-^ 
** 

#* 

$*** 

/Art.  419.] 
r 

Day  Book,  115-797. 
Messrs.  EDWARDS  &  Co., 


BILLS. 

(6.  Groceries.) 


159 


NEW  YORK,  Feb.  1, 1889. 
Bought  of  H.  K.  &  F.  B.  THURBER  &  Co. 


M  14385 

1 

Cask  Old  Prunes  1544  -  134  = 

****  Ibs.  - 

** 

** 

3 

Boxes  Old  Muscatel  Raisins  - 

_    _    _ 

, 

* 

•*# 

3 

"      New        "            " 

_    _ 

QIO 

* 

** 

4 

"     Layer           "        - 

_    _ 

1»6 

* 

** 

1 

"      Cream  Tartar,  £  foil    - 

-  20  Ibs.  - 

.39 

* 

** 

2 

"      Yeast-Cakes,  3  doz.  ea., 

-  6  doz.  - 

.65 

• 

** 

25 

IT,             "\7_7V*  j-vl  rt      7}r\v\wrii« 

.16 

* 

10 

"    Nutmegs  fll       -    -    -    - 

_    _    _    . 

100 

## 

1 

Box  O.  K.  Mustard,  1's    -    - 

-  12  Ibs.  - 

.25 

* 

1 

i's    -    - 

-12    "    - 

.25 

* 

Cartage 

on  all  -    - 

1 

**# 

** 

1st  item — "M  $4385"  is  mark  and  number  upon  the  cask  ;  1544,  gross 
wt. ;  134,  tare  or  weight  of  cask.  5th  item — \  foil,  put  up  in  £  Ib.  packages 
and  wrapped  in  tin  foil. 


(7.  Groceries.) 
Messrs.  HORTON,  CRARY  &  Co., 


NEW  YORK,  Aug.  13,  1886. 
Bought  of  AUSTIN,  NICHOLS  &  Co. 


W.  B 

1 

100       00 

QA 

56 

A*C99 

A. 

1 

«         -20              K 

______ 

Ld&        fwO 

131    21|- 

ou 

** 

** 

1 

Bbl.'85  Roa.  Java  Coffee             ^  - 

100    25| 

** 

** 

2 

"    -so     «    Ri0 

u      112—22    221 

109—20      42  " 

***    24 

** 

** 

H.  R.  P. 

1 

Case  Cone.  Lye    - 

_ 

5 

50 

Union. 

2 

Boxes  Yeast  Cakes, 

ea.  3   -    -    -    - 

*    65 

* 

X* 

25 

Ibs.  Spice,             B 

Bcr  900 

-IKl 

* 

•*# 

-     10^ 

5 

214  26 

* 

** 

A.  N.  &  Co. 

1 

Keg  Gr.  Mustard 

______ 

50    35 

#-* 

** 

10 

lh«    \^hifp  f-i-lnp 

40 

* 

•iv-S" 

J.I/ 

257—20  #*** 

269—20      ** 

A.  N.  &  Co. 

5 

Bbls.  X.  C.  Sugar 

_     .256—21  
253—18 

253—20 

^••SHt1^       11^ 

&#& 

** 

8134 

1 

"      W.  D.  Syrup 

47 

***  60^ 

** 

** 

1114 

1 

"      C.  D.        " 

45* 

***  50 

** 

** 

Ctg.^- 

-    - 

1 

50 

$*** 

XX 

The  small  figures  at  the  right  of  the  words  "  bag "  and  "  bbl"  are  the 
prices  of  the  same.  3rd  item — 121  Ibs.,  gross  wt.,  21  Ibs.  tare,  100  Ibs.  net  wt. 
4th  item— 112  and  109,  gross  weights;  22  and  20,  tare;  221,  total  gross  weight; 
42,  total  tare.  12th  item — ^,  £  gallon  allowance  for  leakage. 


160 


PERCENTA  GE. 


[Art.  419. 


(8.  Dry  Goods.) 

NEW  YORK,  March  20,  1889. 
Messrs.  MARSHAL  FIELD  &  Co., 

Sought  of  H.  B.  CLAFLIN  &  Co. 

Terms  Cash  in  30  days  less  5#,  or  4  months'  note  delivered  within  30  days,  and  payable  at 
Bank  in  New  York  exchange. 


2875 
8039 
3369 
1290 
1590 


2179 
2507 
6515 
2985 
1650 


Bale  Boott  M.  Brown 
"     Continental  C.  do. 
"    Pequot  A.  36  in. 
"    Great  Falls  E.    - 
"    Atlantic  H.    -    - 


-    1038    -.073 


"    Boott  F.  F. 

"     Pepperell  600  Drill       -    -    - 

Case  Blaokstone  A.  A.    -    -    -    - 

"     Dwight  Anchor  ----- 

"    Great  Falls  Q. 

"     Pearl  River  Ticking    -    -    - 
Cooperage 


800 

800 

967 

1111 

** 


800 

622 

1649 

1139 

1492 

708 


71 


8 
152 


54 

** 


-><•# 

#•* 

*#* 

*** 

#** 


** 
** 


How 


L  would  settle  the  above  bill  April  19, 1889  ? 


(9.  Dry  Goods.) 

NEW  YORK,  March  S3,  1888. 
Messrs.  DAVIDGE,  LANDFIELD  &  Co., 

Bought  of  TEFFT,  WELLER  &  Co. 


yd. 

c. 

2 

Naumkeag  Bl.  Jean    -    -    47   -    -    - 

95 

9 

8 

55 

4 

Roll  Cambric    -    -    -    -   J5»  J5  -    " 

**** 

52 

* 

** 

3 

473 
Pepperell  Drill  -    -    -    -    ao3  -    -    - 

#*#* 

8 

* 

** 

443 

1 

T  ir*  \i7oll    1  0  /       "Rtvuvn 

38 

142 

* 

** 

40 

Q 

*** 

72 

* 

o 

40 

1 

453  45 

5 

New  Market  N.       -    -    -    45*  58l      - 

**** 

61 

** 

** 

462 

2 

*** 

9 

* 

** 

2 

Otis  B.  B.  Dk  Stripe  -.   -  Jjf  -  •  -  .  . 

*** 

10 

** 

** 

1 

Hamilton  30  in.  Tick 

483 

II2 

* 

** 

2 

Thorndyke  C     ----583--- 

**** 

82 

*# 

** 

2 

Wamsutta  C.  Blea.      -    -    jjgl  -    -    - 

**** 

12 

** 

** 

8 

Anr1rn«T,                             52     52     49     51' 

Anaros  LI.    •          -   512  513  51   522 

*** 

73 

** 

** 

1 

Pepperell  10/4  

363 

22 

* 

** 

*** 

** 

1st  item  —  2  pieces  Naumkeag  Bleached  Jean  containing  48  and  47  yards 

respectively ;  total,  95  yards  at  9  cents  per  yard. 


Art.  419.] 


BILLS. 


161 


(10.  Dry  G-oods.) 

Book  174,  Page  14$.  NEW  YORK,  March  30,  188S. 

Mr.  JAMES  MORGAN,  Milwaukee,  Wis. 

Sought  of  H.  B.  CLAFLIN  &  Co. 

Terms  :    Net  6O  Days,  or  \%  discount  in  30  days,  or  2#  ) 
discount  in  10  days,  N.  Y.  Funds.    No  Exchange  allowed,  f 


$4641 

53 

PC'S  Gordon  Prints  (Job) 

212  482  38    401  482  483  37*  48    48 

44    492  443  482  492  493  492  42    56 

482  491  282  491  49    483  491  28    48s 

37    33*  492  52    333  40    48    491  491 

24   482  482  52    483  49    472  481  482 

491  492  483  482  482  432  491  49*    - 

***** 

S2601 

54 

PC'S    Do. 

483  48    49    42    221  491  49    482  532 

4g2  473  48a  482  49    44    49    492  432 

492  49    49    482  473  47    482  491  56 

502  491  411  481  50    271  49    482  483 

213  291  513  463  482  482  282  482  491 

492  452  47    482  402  501  392  482  461 

***** 

$4765 

61 

PC'S    Do. 

302  492  42    492  32    48    46    482  462 

423  472  221  33    46    48    492  482  48 

42    42    48    28    481  492  482  49    49 

492  482  282  492  43    491  482  492  48 

382  29    25    263  491  493  491  49    482 

343  433  45    49    491  492  431  36    4g 

292  493  482  311  482  49    481      -    - 

***** 

***** 

.042 

*** 

** 

How  much  would  settle  the  above  bill  April  8,  1888?    How  much  April 

28,  1888? 

(11.  Dry  G-oods.) 

NEW  YORK,  March  20,  1888. 
Messrs.  JORDAN,  MARSH  &  Co. 

Bought  of  A.  T.  STEWART  &  Co. 


Job. 

8 

Cases  Gordon  Fancy 

J.  U. 

S  4561         2810 

S.  B.  R. 

4157        29021 

H.  Z. 

3473        278T2 

S.  J.  L. 

4224        28802 

G.  Q. 

2777        28211 

J.B. 

3504        28422 

J.  Z. 

3970        28831 

J.  H. 

4198        28631     -    -    ******  .05 

**** 

** 

Less  §%     - 

** 

** 

**** 

** 

1st  column,  distinguishing  number  of  each  case.     2d  column,  number  of 
yards  in  the  several  cases. 


162  PERCENTAGE.  [Art.  419, 

(12.  Hosiery.) 

Claims  for  Damages  or  Errors  must 

be  made  on  receipt  of  Goods.  ^EW  YORK,   June  28,   1880. 

Messrs.  JOHN  FORD,  SONS  &  Co., 

Sought  of  JAMES  TALCOTT. 
Net  SO  Days.— Note  to  your  own  order  payable  at  a  Bank  in  New  York  City. 


1789 

35 

Doz.  3458  Mixed  >£  Hose      -    -     .80 

28 

25 

2032  Fancy    "            -    -    -     .80 

** 

12 

853  Col'd                  -    -    -  1.00 

** 

12 

1691  Fancy                 -    -    -  1.00 

#* 

18 

1759      "                     ...    .75 

** 

** 

20 

1713      "                     ...  i.oo 

*# 

16 

1716      "                     -    -    -  1.10 

** 

** 

6 

3438  Fch.  mx.  %       ...    .90 

* 

** 

22 

Job    Misses                -    -    -    .75 

** 

** 

|*** 

** 

Shipped  per  P.R.E.  &  C.B.  &  Q.R.R. 

Number  on  margin  (1789),  number  of  case.     Numbers  3458,  2032,  eie.t 
manufacturer's  distinguishing  numbers  (stock  numbers). 


Mr.  JOHN  BERWOLD, 
Terms  Cash. 


(13.  Books.) 

CHICAGO,  ILL.,  May  7,  1878. 
Sought  of  HADLEY  BROS. 


12 
18 
24 
36 

Randall's  Arithmetics,  Part  1      -     .60 
"     2      -     .50 
Smith's  Primers  (paper)      -    -    -    .06 

Qr»plW<5                                                99 

7 
* 

* 

• 

20 

** 

** 

18 
12 

A 

2d  Readers    .45 

3d        "         -----    .70 

4th        "                                       11^ 

* 
* 
* 

** 

** 

•i'3- 

6 
6 

5th      "         .....  1.35 
Doz.  Brown's  Copy  Books  -    -    -  1.80 

* 
** 

** 
** 

Less  m%% 

** 
** 

** 
** 

** 

•*< 

6 
6 
6 
6 

Jones'  Geographies  *  1    -    -    -    -    .35 
2    -    -    -    -    .63 
"               "            3    -    -    -    -  1.10 
4    -    -    -    -  2.00 

2 

* 
* 
** 

10 
** 

** 

Less  25^      - 

** 
* 

** 
** 

** 

** 

3 
3 

Boxes  Chalk  Crayons     -    -    -    -    .18 
Doz.  Blank  Copy  Books      -    •     -    .50 

* 

** 
** 

$** 

** 

Art.  419.] 


BILLS. 


163 


(14.  Hardware.) 
PHILADELPHIA,  PA.,  Aug.  IS,  1889. 
Messrs.  N.  RUTTER,  Sox  &  Co., 
Bought  of  BIDDLE  HARDWARE  Co. 
Terms  60  days. 

24 

Sets  Wd  Wh'l  Bed  Casters  ft  1  2  in.  -     .18 

* 

** 

50%     -    - 

* 

** 

1 

Doz.  Russell's  S.B.Knives  14  in.  ft  1540    - 

11 

2.40     2.55      3.15      3.20 

200 

Carriage  Bolts  %  x  1        2^    5%    5^ 

«* 

*# 

5.95      6.25      6.50      6.85 

100 

** 

** 

7.15     7.45 

100 

** 

** 

7.90                 8.05 

100 

"     %  x  8^              8%    -    - 

** 

** 

7.25     7.75      9.25 

100 

"     M  x  2        2J£    4  -    -    - 

*# 

** 

11.35    11.75    13.25 

100 

"     ^  x  6        6J£    8  -    -    - 

** 

** 

75  &  12i£%     - 

*** 

** 

M 

** 

Y2 

C.  Machine  Bolts  %  x    8                   8.70 

* 

** 

15.10       16.60 

Y* 

"     %  x     6           7  -    -    - 

** 

*«• 

99 

Ibs.     "          .  "     %  x  11                  .10% 

** 

** 

** 

** 

60  &  10$     - 

** 

** 

** 

** 

3rd  item — 200  bolts  of  each  of  the  following  sizes:  \  in.  thick  x  1  in.  long, 
£  in.  thick  x  2£  long,  £  Vit.  thick  x  5£  long,  ^  *n.  thick  x  5^  in.  long.  The 
numbers  2.40,  2.55,  3.15,  and  3.20  represent  the  prices  per  hundred  of  the 
several  sizes. 

(15.  Watches  and  Jewelry.) 

NEW  YORK,  Mar.  7,  1887. 
Mr.  CHARLES  BABCOCK, 

Bought  of  A.  S.  GARDNER  &  Co. 
Terms:  Net  Cash  4  months,  or  less  5#  30  days,  with  Exchange  on  New  York. 


H  658 

1 

18  k.  Ancre  full  Engrd.  &Enld.  S.  W. 

90 

20422 

1 

14  k.  Russell  flat  C.  B. 

46 

50 

1 

18  k.  Plain  Ring  3%  dwts.     -    -    -      1^ 

* 

#* 

2 

14  k.  Guards  with  slides  ?^,   —     -      l1^- 

222    208 

#*# 

#•» 

1 

Pr.  Solid  Roman  SI.  Buttons  908    -    - 

10 

50 

**# 

*i* 

1 

i        How  much  would  settle  the  above  bill,  Apr.  2,  1887? 

The  letters  and  numbers  on  the  margin  refer  to  the  numbers  of  tho 
watches.  4th  item — numbers  222  and  208  refer  to  the  style  numbers  (stock- 
numbers)  of  the  guards  (chains),  and  the  numbers  above  (37f  and -56)  express 
the  weights  in  pennyweights  ;  $1.15  per  pennyweight. 


164 


PERCENT  A  GE. 


[Art.  419 


(16.  Tinware.) 

ROCHESTER,  N.  Y.,  Oct.  16,  1889. 
Messrs.  MCCARTHY  &  REDFIELD, 

Bought  of  JOHN  H.  HILL. 
Terms  60  days.    If  paid  in  10  days  2  per  cent,  discount. 


2 

Doz.  ft  21  Pieced  Dish  Pans    -    -    8.25 

** 

** 

% 

"    9  in.  Wash  Boilers      -    -    -  36.00 

** 

3 

"    Pieced  Bread  Pans  3x9x3-    2.00 

# 

3 

"5x9x2-    2.00 

* 

3 

"    ft  13  Pieced  Cups    -    -    -    -      .90 

* 

## 

2 

"    ft  25  Dippers            ...    -     1.75 

* 

** 

6 

Nests  ft  021  Flaring  Pis  &  Dippers    1  ,14 

* 

** 

** 

** 

20  &  12%$     - 

•*# 

** 

** 

** 

1 

Doz.  Champion  Nutmeg  Graters 

1 

75 

1 

"    Nests  ft  4  Fancy  Cov'd  Pails 

6 

00 

1 

"    ft  4  Burnished  Tea  Pots  -    - 

6 

75 

85  ft  UK*     - 

#•* 

** 

** 

2 

Doz.  ft  10  Pudding  Pans    -    -    -    4.25 
"    ft  200  Pressed  Kettles      -    -    5.50 

* 
* 

** 

37%%     - 

* 

** 
** 

» 

** 

.76    .90 

6 

Enameled  Kettles  Ea.  4—5  qt.  - 

* 

** 

1.10  1.30 

12 

"    6—8  qt.  - 

** 

** 

60%     - 

** 

#* 

** 

** 

** 

N.Y.C.  &  H.R.R.R.  975  Ibs.  @  \W 

-X-* 

** 

What  amount  would  be  due  on  the  above  bill  Oct.  26,  1889? 

(17.  Wooden  "Ware.) 

CHICAGO,  July.  9,  1887. 
Messrs.  OLIVER  &  BACON, 

Bought  of  JAMES  S.  BARRON  &  Co. 

Terms  Cash,  with  exchange  on  Chicago  or  New  York. 

6 

Oak  Churns  ft  1     1.60 

* 

** 

Less  20%     - 

* 

#* 

* 

** 

2 

Doz.  1  bu.  Corn  Baskets    -    -    -    4.00 

* 

¥ 

"     Potato  Mashers      -    -    -    -    1.00 
"    6  ft.  Ladders     

4 

•»* 

1 

"8ft.        "          

5 

•4 

«    jo  ft       " 

/> 
™ 

** 

Less  50  &  10%     - 

* 

** 

* 

** 

2 

"    ft  10  Shoe  Brushes                 -    2.25 

* 

** 

** 

** 

Art.  419.]  BILLS.  ,         '16o 

NOTE. — In  the  preparation  of  bills  in  the  usual  form,  from  the  following 
items,  use  your  own  name  as  the  seller,  the  name  of  your  teacher  as  the  buyer, 
and  the  present  date  and  place. 


'18.  \gro.  Table  Knives  and  Forks  @  $8.40.  TV  doz.  Cheese 
Knives  @  $9.60.  |  doz.  Razors  each  #100  $9,  #101  $10,  #102 
$10.50.  \  doz.  Pocket  Knives  each  #337  $6,  #427  $7.50,  #204 
$3.75.  6  sets  Champion  Irons  @  $1.50.  1  doz.  Tacks  each  #1 
2#6'.,  #2  22c.9  #3  25c.,  #4  21c.  f  doz.  Panel  Saws  @  $20  less 
20$.  Box  and  drayage,  75<?.  Terms:  Cash.  (See  Ex.  14.) 

19.  2293  Us.  S.  Meal  @  $14  ^  per  cwt.     200  III.  Dakota  Flour 
@  $7_2A.     H890  II.  Feed  @  $1^.     170  Bags  (To  be  returned) 
@  25c.     Expressage  on   Empty  Bags,  25c.     Car  #30808.     (See 
Ex.  2.) 

20.  2  doz.  Wrought  Butts  each  3|-  x  3  $2.40,  3|  x  4  $3,  less  55 
and  10$.     H   doz.  Locks  #184  @  $48  less  30$.     1  ^oz.   Locks 
#476  $15  ;  1  doz.  Knobs  #700  $6.50  ;  1  doz.  Escutcheons  #16  $2, 
less  45  and  5$.     (On  last  three  items.)     3  doz.  Sash  Fasteners 
#15  @  $2.25.     4  doz.  Solid  Eye  Mattocks  @  $15.50,  less  33^(1)$. 
Box  and  cartage,  63c.     Terms:  60  days.    (See  Ex.  14.) 

21.  2  doz.  Smith's  Bitters  @  $7.25.     110  II.  Epsom  Salts  @ 
3{c.     2  doz.  Sweet  Oil  #3  @  $1.75.     2  doz.  Sweet  Oil  #4  @  $1.25. 
2  ^02.  Paregoric  2  oz.  @  $1.25.     2  6?oz.  Laudanum  2  02.  @  $2.25. 
2  do£.  S.  M.  Oil  @  $1.25.     2  dos.  Extract  Lemon  2  00.  @  $2.     2 
6/02.  Brown's  Syrup  @  $1.75.     2  6?0£.  Fancy  Soap  @  67c.     2  ^02;. 
Castor  Oil  @  $1.75.     2  ^o^.  Golden  Liniment  @  $1.85.     Cartage 
506'.     Terms:  Net  Cash,  without  discount. 

S22.  2  III.  Prunes  248  —  20,  285  —  20,  @  6c.  2  Z>£/.  Eice 
40  —  19,  229  —  20,  @  6|c.  5  iJ7.  "A"  Sugar  319,  306,  288,  319, 
306  II.  @  9|c.  5  JJ/.  Yellow  C  Sugar  314-19,  319-20,  329  —  20, 
311  —  21,  328—19,  @  8}c.  1  W?.  Cut  Loaf  Sugar  236-20  @ 
10|c.  Cartage,  $1.25.  (See  Ex.  7,  llth  item.) 

28.  8  lengths  8"  Drive  Pipe  129'  10"  @  $2,25  (per  foot) ;  44 
lengths  5|"  Casing  801'  8"  @  70c.;  233  lengths  2"  0.  W.  Tubing 
4507'  @  216-.,  less  7J-  and  2$.  Less  freight  29400  ^.  @  236'.  per  cwt. 

24.  12  Pr.  Women's  Grain  B't.  #443  Shoes  @  $1.25.  12  Pr. 
TTos.  Kid  B't.  #407  @  $1.50.  12  Pr.  Wos.  Kid  P7.  #406  @ 
$1.75.  12  Pr.  Misses  Kid  B't.  #301  @  $1.50.  12  Pr.  Misses 
Goat  B't.  #302  @  $1.60.  12  Pr.  Wos.  Goat  5V.  #428  @  $1.75. 
12  Pr.  Children's  Goat  PV.  #200  @  $1.30.  12  Pr.  Oh.  Grain 
57.  #202  @  $1.20.  12  Pr.  Oh.  Glove  Kid  57.  #222  @  $1.10. 


1G6  PERCENTAGE.  [Art.  42  O. 


COMMISSION   AND   BROKERAGE. 

420.  Commission  or  Brokerage  is  an  allowance  made  to 
an  agent  for  transacting  business  for  another ;   as,  the  sale  or 
purchase  of  property,  the  collection  or  investment  of  money,  etc. 

An  additional  percentage  is  usually  charged  by  commission  merchants  for 
guaranteeing  the  payment  of  sales  made  on  credit. 

421.  The  party  who  transacts  the  business  is  called  a  Com- 
mission Merchant,  or  Broker ;  and  the  one  for  whom  he  acts 
is  called  the  Principal. 

NOTES. — 1.  Commission  Merchants  usually  have  possession  of  the  subject- 
matter  of  the  negotiation,  and  make  sales  and  purchases  in  their  own  name. 

2.  Brokers  do  not  have  possession  of  the  merchandise  bought  or  sold,  and 
generally  make  contracts  in  the  name  of  those  who  employ  them  and  not  in 
their  own.  They  simply  effect  bargains  and  contracts. 

The  name  broker  is  often  erroneously  applied  to  dealers  in  stocks,  bonds, 
etc.,  who  buy  and  sell  on  their  own  account  only. 

4:22.  A  Consignment  is  a  quantity  of  merchandise  sent  by 
one  party  to  another.  The  party  who  sends  it  is  called  the  Con- 
signor; and  the  party  to  whom  it  is  sent,  the  Consignee. 

4:23.  The  Net  Proceeds  of  a  consignment  is  the  balance  due 
the  consignor  after  all  charges  or  expenses  have  been  deducted. 

The  whole  amount  realized  from  a  sale  is  called  the  gross  proceeds.  Th-- 
commission  is  usually  a  certain  per  cent,  of  this  amount. 

424.  An  Account  Sales  is  a  detailed  statement  rendered 
by  the  Commission  Merchant  to  the  Consignor,  showing  the  sales 
of  certain  goods,  the  charges  or  expenses  attending  the  same,  and 
the  difference  or  net  proceeds. 

The  charges  embrace  freight,  cartage,  inspection,  advertising,  storage, 
insurance,  commission  and  guarantee,  etc. 

425.  An  Account  Purchase  is  a  detailed  statement  rendered 
by  the  Commission  Merchant  to  his  Principal,  showing  the  cost 
of  certain  goods  bought,  and  the  charges  or  expenses  attending 
the  purchase. 

426.  Commission  or  brokerage  is  usually  computed  afc  a  cer- 
tain per  cent,  of  the  amount  realized  or  invested,  or  of  the  amount 


Art.  427.]       COMMISSION    AND     BROKERAGE.  107 

involved  in  the  transaction.     In  such  cases  the  general  principles 
of  percentage  are  applied. 

NOTES. — 1.  In  buying  and  selling  stocks,  bonds,  etc.,  the  par  value,  and 
not  the  actual  value,  is  taken  as  the  base. 

2.  The  commission  for  buying  and  selling  some  kinds  of  merchandise  is 
usually  computed  at  a  certain  price  per  unit  of  weight  or  measurement  ;  as, 
grain  per  bushel,  cotton  per  bale,  etc. 

EXAMPLES. 

427.  1.  A  commission  merchant  sold  goods  to  the  amount 
of  $864  ;  what  was  his  commission  at  2 £  (J-  of  10)  %  ? 

2.  A  salesman  sells  goods  at  a  commission  of  2£%  ;  what  must 
be  his  sales,  that  he  may  have  a  yearly  income  of  $5000  ? 

3.  What  is  the  brokerage  for  selling  850  bales  of  cotton  at  the 
rate  of  $25  per  100  bales  ? 

4.  A  lawyer  collected  a  note  of  $2375 ;  how  much  did  he  pay 
to  the  owner  of  the  note,  his  commission  being  5%  ? 

5.  My  agent  in  Chicago  purchases  for  me  600  barrels  of  flour 
lit  $3.75  per  barrel ;  how  much  do  I  owe  him,  his  commission  for 
purchasing  being  2%  ? 

6.  An  officer  collected  $17850,  and  deposited  $17493  in  the 
Treasury,  retaining  the  remainder  as  his  commission.     What  was 
the  rate  per  cent,  of  the  commission  ? 

7.  Sent  to  a  commission  merchant  in  Toledo  $2080.80  to  in- 
vest in  flour,  his  commission  being  2%  on  the  amount  expended ; 
how  many  barrels  of  flour  would  be  purchased  at  $4.25  per  barrel  ? 

8.  A  commission  merchant  sells  merchandise  amounting  to 
$3325 ;  how  much  is  paid  to  the  consignor  of  the  merchandise, 
the  charges  being,  for  transportation  $117.50,  for  advertising  $10, 
for  storage  $15,  for  commission  2£%  ? 

9.  My  agent  in  Chicago  buys  for  me  1187.76  centals  wheat  at 
$2.123  per  cental.     What  is  his  commission  at  £  per  cent.  ? 

10.  A  commission  merchant  purchased  for  me  9-S-&  bushels  of 
clover  seed  at  $8.55  per  bushel.     How  much  should  I  send  to  him 
in  settlement,  if  his  commission  for  purchasing  is  1  per  cent.  ? 

11.  A  broker  buys  8375  pounds  of  leather  at  26  cents  per 
pound.    What  is  his  brokerage  at  j-%,  and  what  is  the  net  amount 
received  by  the  seller,  the  brokerage  being  paid  by  him  ? 

12.  A  freight  broker  procures  transportation  for  375  tons  of 
merchandise  at  $3.50  per  ton  ;  what  is  his  brokerage  at  $%  ? 


1G8  PERCENTAGE.  [Art.  427. 

13.  A  collector  deposits  $28117,   retaining  3%  on  the  whole 
amount  collected.     What  amount  did  he  collect  and  what  was  his 
commission  ? 

14.  A  lawyer,  collecting  a  note  at  a  commission  of  5%  thereon, 
received  $6.25  ;  what  was  the  face  of  the  note  ? 

15.  An  agent  sold  6  mowing-machines  at  $120  each,  and  12  at 
$140  each.     He  paid  for  transportation  $72,  and,  after  deducting 
his  commission,  remitted  $2208  to  the  manufacturer.     What  was 
the  %  of  his  commission  ? 

16.  A  merchant  instructs  his  agent  in  Cincinnati  to  huy  pork 
to  the  amount  of  $5000.     The  charges  on  the  pork  being  $16,  and 
the  agent's  commission  1£$,  how  much  must  be  remitted  to  settle 
the  bill  ? 

17.  What  are  the  net  proceeds  of  the  sale  of  12372  pounds  of 
leather  at  22  cents  per  pound,  the  charges  being  $31,  and  a  com- 
mission of  2J$  being  paid  for  selling  and  2J%  for  guaranteeing 
payment  ? 

18.  A  real  estate  agent,  who  charged  2J%  for  making  the  sale, 
paid  to  the  owner  of  a  house  and  lot  $42412.50;  what  was  the 
value  of  the  property  ? 

19.  A  commission  merchant  sells  240  bbl.  of  potatoes  at  $3.75 
per  1)1)1.,  and  260  bbl.  at  $3.60  per  libl.     How  much  is  due  the  con- 
signor, the  commission  being  12J  cents  per  barrel  ? 

20.  John  Smith  is  a  disbursing  agent  of  the  United  States. 
Jan,  1,  1880,  there  is  in  his  hands  $11870.63.     Feb.  1,  he  pays  out 
$3220.34,  on  which  he  is  entitled  to  a  commission  of  \\%.     Mar. 
1,  he  receives  $3750.87.     May  1,  he  pays  out  $3795.01,  on  which 
he  is  entitled  to  a  commission  of  2J%.     Make  a  statement  of  his 
account,  showing  balance  due  the  United  States. 

21.  A  lawyer  collected  75%  of  an  account  of  $3416,  charging 
5%  commission.     What  amount  should  he  pay  over  ? 

22.  A,  having  a  claim   against   the   government   of   $10970, 
agreed  to  pay  an  agent  8  per  cent,  of  the  amount  collected.     The 
amount  collected  was  22  per  cent,  less  than  the  amount  of  the 
claim.     How  much  was  received  by  A  ? 

23.  B  sends  $2240. 70  to  his  agent  in  Cleveland,  requesting  him 
to  invest  in  provisions  after  deducting  his  commission  of  %%  for 
purchasing ;  what  was  the  sum  invested  ? 

2Jf..  A  broker  received  $62.50  for  selling  some  bonds,  charging 
\%  brokerage.     What  was  the  par  value  of  the  bonds  ? 


Art.  427.]        COMMISSION    AND     BROKERAGE. 


169 


Copy  the  following  account,  and  make  the  necessary  exten- 
sions, etc. 

(25.  Account  Sales.) 

NEW  YORK,  Oct.  19,  1889. 
Sold  for  account  of  A.  W.  RANDOLPH  &  Co., 

By  DAVID  Dows  &  Co. 


1880. 
Sept. 
(i 

H 

Oct. 
t< 

12 
18 
30 

14 

18 

100  Bbls.  "Sunshine"-    -    -    -    5.75 
125     "      "  Pride  of  the  West  "-    6.25 
150     "      "Sunshine"-    -    -    -    6. 
75     "      '  'Pride  of  the  West  "-    6.50 
50     "                "                "        -    6.60 

*** 
*** 
#** 
*** 
*** 

** 
** 

**** 

** 

Sept. 
Oct. 

10 
10 
19 

1Q 

Charges. 
Transportation  500  Bbls.  @  27^  -    -    - 
Cartage              400     "      @    5^-    -    - 
Storage              400     "      @    3j*-    -    - 

*** 
** 
** 
# 

** 

« 

19 

Commission  and  Guarantee  5%  -    -    - 

*## 

** 

*** 

** 

Net  proceeds      

##** 

** 

26.  According  to  the  above  form,  prepare  an  Account  Sales 
of  10  III.  Yellow  C  Sugar,   3031  Ib.  ®  8c.;  10  bbl  Standard  A 
Sugar,  2957  Ib.  @  9^.;  10  bbl  Soft  A  Sugar,  2839  Ib.  @  8|c.; 
2   Tc.   Lard,   713  Ib.  @  9|-c.;  1   Tc.  Rice,  608  Ib.  @  Ic.     Charges 
as  follows  :    Cooperage,  $1.80  ;  Cartage,  $3.60  ;  Commission,  1J^. 
Present  date  and  place ;  Student  &  Co.,  commission  merchants ; 
and  sold  for  account  of  Teacher  &  Co. 

27.  If  an  agent's  commission  is  $145.20,  when  he  sells  $5808 
worth  of  goods,   how  much  would  it   be   when   he   sells  $7416 
worth  ? 

28.  A  creditor  receives  on  a  debt  of  $1725,  a  dividend  of  60%, 
on  which  he  allows  his  attorney  b%.     He  receives  a  further  divi- 
dend of  25%,  on  which  he  allows  his  attorney  6%.     What  is  the 
net  amount  that  he  receives  ? 

29.  A  gentleman  left  a  sum  of  money  to  be  divided  equally 
among  7  persons,   subject  to  an  inheritance  tax  of  5%,  which 
caused  a  deduction  of  $364  from  the  whole  amount.     What  did 
each  receive  ? 

30.  An  agent's  commission  for  the  month  was  $128.40.     If  his 
sales  had  been  $864  more,  his  commission  would  have  been  $150. 
Find  the  amount  of  his  sales. 

31.  A  man  allows  his   agent   5%  on   his   gross   rentals,  and 
receives  a  net  rental  of  $3488.40.     If  the  gross  rental  is  6%  of  the 
value  of  the  property,  what  is  the  value  of  the  property  ? 


^  c  •&* 


PERCENTAGE.  [Art.  427. 

(32.  Account  Purchase.) 
i 

TOLEDO,  0.,  Mar.  6,  ltf$7. 
Purchased  by  A.  L.  BACKUS  &  SONS, 

For  account  and  risk  of  L.  A.  &  W.  B.  SHAW. 


9 

227 
9JJL 

9JLL 

Bags  "Montauk"     .21 
Bu.  Mammoth  Clover  Seed     -    -    99— 
"    Clover  Seed    8^ 

* 
** 
** 

** 
#* 

** 
25 

*»„ 

H 

Charges. 

* 

Commission  \%    ------ 
Charge  your  %     ------ 

*** 

** 

NOTE. — The  small  figures  at  the  left  represent  pounds.     See  Art.  »53S. 

33.  According  to  the  above  form,  prepare  an  Account  Purchase 
of  3  Half-Chests  Gunpowder  Tea,  165  Ib.  @  35c.;  2  Hfc.  Oolong 
Tea,  86  Ib.  @  20e.;  20  bags  Rio  Coffee,  2388  Ib.  @  13c.;  2  mats 
Java  Coffee,  133  Ib.  @  19|-c.;  1  Hhd.  P.  R.  Molasses,  143  yal  (<r, 
54c.  Charges  as  follows:  Drayage,  $1.75;  Commission,  1 ',. 
Commission  Merchants,  Student  &  Co.;  bought  for  Teacher  & 
Co. ;  present  place  and  date. 

34-  After  paying  an  auctioneer  5$,  a  man  received  $1172.30 
for  his  furniture.  What  were  the  gross  sales  ? 

35.  A  bankrupt's  assets  are  $17415,  and  his  liabilities,  $4837"). 
I  place  my  claim  of  $2560  in  the  hands  of  my  attorney  for  collec- 
tion.    How  much  do  I  receive  if  the  attorney  retains  5^  commis- 
sion ? 

36.  A  landlord   received  $822  as  the  net  rental  of  a  house, 
after  his  agent  had  paid  $60  for  repairs  and  charged  2%  commis- 
sion on  the  gross  rental.     What  was  the  gross  rental  ? 

37.  C  of  New  York  sells  for  D  of  Atlanta,  a  quantity  of  cotton, 
amounting  to  $7317.83,  and  charges  a  commission  of  %\%.     By 
instructions,  he  invests  the  proceeds  in  dry  goods,  after  deducting 
a  commission  of  \\%  on  the  amount  expended.     What  was  the 
total  commission  ? 

38.  A  commission  merchant  sold  300  bales  of  cotton,  averaging 
462  Ib.  to  the  bale,  at  15. 7^,  his  commission  being  25^  per  bale, 
and  the  charges  $161.     He  purchased  for  the  consignor  dry  goods 
amounting  to  $2576.37,  charging  a  commission   of   \\%.     How 
much  was  still  due  the  consignor  ? 


INTEREST. 


428.  Interest  is  a  sum  charged  for  the  use  of  money,  or  its 
equivalent ;  or  more  strictly  speaking,  it  is  the  use  of  money,  or 
the  service  rendered  in  its  use. 

439.  The  Principal  is  the  sum  for  the  use  of  which  interest 
is  charged. 

430.  The  Hate  is  the  per  cent.,  or  number  of  hundredths, 
of  the  principal,  charged  for  its  use  for  a  certain  time,  usually  for 
one  year  (per  annum).    When  no  time  is  mentioned  with  the  rate 
in  the  contract,  a  year  is  understood. 

431.  The  Amount  is  the  sum  of  the  principal  and  interest. 

If  $1000  is  loaned  for  one  year  at  6%  per  annum,  $60  would  be  the  inter- 
est, $1000  the  principal,  and  $1060  the  amount. 

432.  Simple  Interest  is  interest  on  the  principal  only  for 
the  full  time. 

433.  Compound  Interest  is  interest  not  only  on  the  prin- 
cipal, but  on  the  interest  also  after  it  becomes  due. 

If  $1000  is  loaned  Jan.  1,  1881,  for  2  years,  the  amount  due  Jan.  1,  1883, 
at  Qfo  simple  mter-est,  would  be  $1000  (Principal)  plus  $120  (Simple  Interest), 
or  $1120.  At  compound  interest  the  amount  due  Jan.  1, 1882,  would  be  $1060 
($1000  +  $60);  the  amount  due  Jan.  1,  1883,  would  be  $1060  plus  $63.60  (§% 
of  $1060),  or  $1123.60.  The  simple  interest  for  2  years  would  be  $120  ;  the 
compound  interest  for  the  same  time,  $123.60.  When  the  word  interest  is 
used  alone,  simple  interest  is  understood. 

434.  Legal  Interest  is  the  interest  according  to  the  rate 
per  cent,  fixed  by  law  for  cases  in  which  the  rate  per  cent,  is  not 
specified.     By  special  agreement  between  parties  in  certain  States, 
interest  may  be  received  at  a  rate  higher  than  the  legal  rate.     In 
most  of  the  States,  this  rate  is  limited.     See  Art.  436. 

435.  Usury  is  the  taking  of  a  higher  rate  of  interest  than 
that  allowed  by  law.     A  person  taking  usury  is  liable  to  certain 
penalties  differing  in  the  several  States. 


172 


INTEREST. 


[Art.  436. 


436.  The  following  table  shows  in  the  first  column  the  legal 
rate  of  interest  when  no  rate  is  specified  in  the  contract,  and  in 
the  second  column  the  maximum  rate  allowed  by  law. 


State  or  Territory. 

Ra 

te. 

State  or  Territory. 

Ra 

te. 

Alabama  

8% 

8% 

Mississippi 

6% 

10% 

aAlaska  (Ter.)  .  .  . 

Missouri 

6% 

10% 

Arkansas 

6% 

10% 

Montana  (Ter  ) 

10% 

Anv 

Arizona  (Ter.).  

10% 

Any 

Nebraska  

7% 

10% 

1% 

Anv 

Nevada  

1% 

Any 

cColorado^.  .  . 

10% 

Any 

New  Hampshire.     .  . 

6% 

6% 

Connecticut 

Q% 

6% 

New  Jersey             .  . 

6% 

6% 

Dakota  (Ter.)  

1% 

12% 

New  Mexico  (Ter.)..  .  . 

6% 

12% 

Delaware  

6% 

6% 

dNew  York  

6% 

6% 

Florida  

8% 

Any 

North  Carolina  

6% 

8% 

Georgia  

7$. 

8% 

Ohio  

6% 

8% 

Idaho  (Ter  ).  . 

10% 

18% 

Oregon     

8% 

10% 

Illinois 

6% 

8% 

Pennsylvania.  .  .  . 

6%  ' 

6% 

Indian  (Ter  ) 

6% 

Any 

Rhode  Inland 

6% 

Any 

Indiana  

6% 

8% 

South  Carolina  

1% 

1% 

Iowa  .  .        .  .         ... 

6% 

10% 

Tennessee  

6% 

6% 

Kansas 

7% 

12% 

Texas  

8% 

12% 

Kentucky.  

6% 

6% 

Utah  (Ter.)  

10% 

Any 

Louisiana 

5% 

8% 

Vermont 

6% 

6% 

Maine  

6% 

Any 

Virginia  

6% 

Q% 

Maryland  

6% 

6% 

Washington  (Ter.).  .  .  . 

10% 

Any 

Massachusetts 

6% 

Any 

West  Virginia  

6% 

0% 

Michigan 

7  % 

10% 

Wisconsin  

1% 

10% 

Minnesota 

1% 

10% 

Wyoming  (Ter.)  

12% 

Any 

(a)  Not  organized. 

(b)  "On  judgments  recovered  in  the  courts  7%,  but  must  not  be  com- 
pounded in  any  manner." 

(c)  "Most  banks  pay  6%  on  time  deposits  and  charge  from  1  to  2%  per 
month  on  loans." 

(d)  "Advances  payable  on  demand  (call  loans),  of  not  less  than  $5000,  on 
negotiable  collaterals,  are  not  subject  to  the  interest  laws,  but  may  be  made 
for  any  compensation  agreed  upon  in  writing." 

437.  Interest  for  Parts  of  a  Year. — Although  many  of  the 
States  have  rigid  laws  in  regard  to  the  rate  per  cent,  to  be  charged 
per  annum,  few  of  them  specify  on  what  basis  interest  should  be 
reckoned  for  a  period  of  time  less  than  a  year.  The  following 
methods  are  in  common  use  : 


Art.43T.]  .,  INTEREST.  173 

1.  Finding  the  time  in  months  and  days  (Compound  Subtrac- 
tion, Art.  31O),  and  regarding  the  months  as  twelfths  of  a  year, 
and  the  days  as  thirtieths  of  a  month  or  360ths  of  a  year.     This 
method,  although  implied  by  the  general  interest  laws*  of  the 
State  of  New  York,   is  not  uniform,  since  it  allows  the  same 
interest  for  February  with  its  28  days  as  for  March  with  its  31 
days.     Its  results  are  sometimes  greater  and  sometimes  less  than 
those  of  accurate  interest. 

2.  Finding  the  exact  time  in  days  (31O)  and  regarding  the 
days  as  360ths  of  a  year.     Since  a  day  is  ^  of  a  year,  this  method 
produces  too  great  a  result.     It  is  used  by  merchants  and  bankers, 
generally,  and  by  many  banks  f  in  discounting  notes.     6%  by  this 
method  is  equivalent  to  §^%%  accurate  interest. 

3.  Accurate    Interest. — Finding   the   exact  time  in  days 
(31O)  and  regarding  the  days  as  365ths  of  a  year.  '  This  method 
is  used  by  the  United  States  government,  and  by  some  merchants 
and  banks ;  but,  on  account  of  its  inconvenience  when  interest 
tables  are  not  used,  it  is  not  generally  adopted. 

NOTES. — 1.  By  the  first  method,  the  time  from  July  10  to  Sept.  10,  would 
be  2  months,  and  the  interest  would  be  T%  or  \  of  the  interest  for  one  year. 
On  $10000  at  6%  for  2  months,  the  interest  would  be  $100  (\  of  .06  of 
$10000). 

2.  By  the  second  method,  the  interval  between  the  same  dates  would  be 
62  days,  and  the  interest  would  be  ^/$  of  the  interest  for  one  year.   On  $10000 
at  6%  for  /^  of  a  year,  the  interest  would  be  $103.33  (^  of  .06  of  $10000). 

3.  By  the  third  method,  the  interval  between  the  same  dates  would  be 
62  days  as  in  the  second  method,  and  the  interest  would  be  -^V2^  °f  the  interest 
for   one   year.     On   $10000  at  §%   for  -£/%  of  a  year,  the  interest  would  be 
$101.92  (3%2F  of  .06  of  $10000). 

4.  The  difference  between  ordinary  interest  and  accurate  interest  for  the 
same  number  of  days  is  -fa  of  the  former,  or  ^  of  the  latter.     Thus  in  the 
above  example,  the  difference  between  the  results,  $1.41  ($103.33-101.92),  is 
^  of  $103.33,  or  ^  of  $101.92. 

*  "  For  the  purpose  of  calculating  interest,  a  month  shall  be  considered  the  twelfth  part 
of  a  year,  and  as  consisting  of  thirty  days;  and  interest  for  any  numher  of  days  less  than  a 
month  shall  he  estimated  by  the  proportion  which  such  number  of  days  shall  bear  to  thirty," 

t  According  to  the  banking  laws  of  the  State  of  New  York,  banks  are  authorized  in 
discounting  notes  to  charge  interest  in  advance  for  the  exact  number  of  days  which  the 
note  has  to  run  (Ch.  XVIII,  Title  2,  §300). 

This  law  appears  to  conflict  with  the  law  quoted  above  which  implies  that  the  time 
shall  be  found  in  months  and  days.  It  does  not  state  whether  the  days  shall  be  regarded  ae 
860the  or  3G5ths  of  a  year. 


174  INTEREST.  [Ari.  438 

438.  Interest  is  an  application  of  percentage,  the  element  of 
time  being  introduced.     Therefore  the  four  elements  or  parts  in 
interest  are  the  Principal  (the  Base),  the  Rate,  the  Interest  (the 
Percentage),  and  the  Time  ;  any  three  of  which  being  given,  the 
other  may  be  found. 

439.  To  find  the  interest  *  for  any  number  of  years 
and  months. 

Ex.    What  is  the  interest  and  amount  of  1324,  for  2  yr.  3  ino., 

at  8^? 

OPERATIONS. 

$324      Principal.  Or  $324 

.08  .18 

25.  92      Interest  for  1  yr.  2592 

2  324 


648  58.32 

5184  324. 


58. 32      Interest  for  2}  yr.  $382. 32 

324  Principal. 


$382. 32      Amount  for  2J  yr. 

ANALYSIS.— At  8#,  the  interest  of  $324  for  1  year  is  .08  of  $324  (the 
Principal),  or  $25.92.  If  the  interest  of  $324  for  1  year  at  8%  is  $25.92,  for 
2  yr.  3  mo.  (2%  yr.),  it  is  2£  times  $25.92,  or  $58.32.  The  amount  is  $324  plus 
$58.32,  or  $382.32. 

44O.  RULE. — To  find  the  interest,  multiply  the  principal 
by  the  rate  per  cent,  expressed  decimally,  and  that  product 
by  the  number  of  years,  and  the  months  as  a  fraction  of  a 
year. 

To  find  the  amount,  add  the  principal  to  the  interest. 

NOTES. — 1.  When  the  rate  per  month  is  given,  apply  the  same  rule,  i.  ?.. 
multiply  the  principal  by  the  rate  per  month  expressed  decimally,  and  that 
product  by  the  number  of  months. 

2.  Instead  of  multiplying  by  the  rate  and  time  separately,  the  process  may- 
lie  shortened  by  multiplying  the  principal  by  the  product  of  the  rate  and  time. 
In  the  above  example,  multiply  $324  by  .18  (2$  x  .08). 


*  Unless  the  words  "Accurate  Interest v  are  used,  all  computations  in  this  book  are  made 
on  the  basis  of  330  days  to  the  year. 


Art.  441.]  INTEREST. 


EXAMPLES. 

441.  Find  the  interest  of 

1.  $875  for  2  yr.  at  1%.  6.  $816.40  for  5  yr.  3  mo.,  at 

2.  $642.50  for  3  yr.  at  §%.  7.  $1275  for  7  yr.  at  6#. 

3.  $1010. 10  for  6  yr.  6  mo.,  at  8$.  5.  $2789.40  for  3  yr.2  mo.,ai 

4.  $3010. 75  for  3  yr.  4  mo.,  at  7$.  9.  $456.75  for  4  yr.  8  mo.,  at 

5.  $3745.80  for4#r.l  rao.,at  6%.  Jf0.  $10180  for  3  yr.  4  wo.,  at 

In  the  following  examples,  find  the  time  by  Compound  Subtraction  (31O). 

11.  What  is  the  interest  of  $6488  from  May  3,  1889,  to  Sept. 
55,  1891,  at  1%  ? 

12.  What  is  the  amount  of  $396.60  from  Aug.   16,  1890,  to 
Dec.  16,  1892,  at  8^  ? 

13.  Compute  the  interest  of  $250.75  from  Nov.  20,  1892,  to 
July  20,  1894,  at  Q%. 

14.  Loaned  on  interest,  New  York,  Dec.  16,  1880,  $1739.75 
(no  rate  specified);  what  amount  should  I  receive,  June  16,  1881  ? 

442.  To  find  the  ordinary  interes     (360  days  to  the 
year)  for  any  rate  and  time.* 

443.     First  or  Day  Method. 

The  interest  of  $1  for  1  year  at  36%  is  $.36. 
"     $1  "    1  day  at  36%  is  $.001. 
"  "     $1  "    I       "        6fo  is  $.OOOJ. 

"     $1  "    1       "        9jg  is  $.OOOJ. 
"     $1  "    1       "        4%is$.000f 
"     $1  "    1       "      12%is$.OOOJ, 
"  "     $1  "    1       "        3%  is  $.000^. 

"     fl  "    1       "        4J%  is  $.000±. 
Ex.     What  is  the  interest  of  $1735  for  73  days  at  6%  ? 

OPERATION. 

$1735 

73  ANALYSIS. — The  interest  of  $1735  for  73  days  is  equiv- 

alent  to  the  interest  of  73  times  $1735,  or  $126655  for  1  day. 

0205  Since  the  interest  of  $1  for  1  day  is  \  of  a  mill,  the  interest 

12145  of  $126655  for  1  day  is  as  many  mills  as  6  is  contained 

6  )  126655  times  in  126655,  or  21109  mills,  or  $21.11. 

$21.109 

*  The  student  should  be  taught  at  least  two  (to  be  selected  by  the  teacher)  of  the  follow- 
ing methods  of  reckoning  interest. 


176  INTEREST.  [Art.  444. 

444.  RULE. — Multiply  the  principal  by  the  nuinber  of 
days,  and  place  the  point  three  places  to  the  left.    The  result 
will  be  the  interest  at  36%.    To  find  the  interest  at  6%,  divide 
by  6 ;  at  4%,  by  9 ;  at  9%,  by  4 ;  at  3%,  by  12 ;  at  12%,  by  3 ; 
at  4t%,  by  8. 

NOTES. — 1.  Observe  in  the  above   rule  that  the  rate  and  divisor,  when 
multiplied  together,  produce  36. 

2.  To  find  the  interest  at  other  rates,  find  it  first  at  6%,  and  then  apply 
the  following  rules  :   At  1$,  divide  by  6  (second  time);  at  1^$>,  divide  by  4  ; 
at  2f0,  divide  by  3  ;  at  5%,  subtract  £  ;  at  7$,  add  £  ;  at  8$,  add  i  ;  at  10$, 
divide  by  6,  and  multiply  by  10  by  placing  the  point  one  place  to  the  right  ; 
at  any  per  cent.,  divide  by  six  (second  time)  and  multiply  by  the  rate. 

3.  If  the  principal  is  a  multiple  of  the  divisor  (6  in  the  model  example), 
time  can  be  saved  by  performing  the  division  first.     Thus,  to  find  the  interest 
of  $1200  for  113  days,  divide  1200  by  6,  and  multiply  the  quotient  200  by  113, 
producing  22600.     By  pointing  off  three  places,  the  required  interest  is  $22.60. 

4.  When  the  time  is  expressed  in  years,  months,  and  days,  reduce  it  to 
days  by  regarding  each  year  as  360  days  and  each  month  as  30  days. 

EXAMPLES. 

445.  Find  the  interest  of 

</.  $1000  for  80  days  at  9$.         6.  $375.60  for  29  days  at  3$. 

$1700  for  77  days  at  6$.          7.  $414.40  for  47  days  at  12$. 
3.  $487  for  33  days  at  4$.  8.  $516  for  95  days  at  5$. 

$1375  for  17  days  at  4J$.       9.  $474  for  19  days  at  7$. 

$2416  for  117  days  at  6$.  \10.  $876  for  83  days  at  8$. 

11.  $387.60  for  3  mo.  17  da.,  at  6$.     At  5$.- 

12.  $1728  for  1  yr.  2  mo.  23  da.,  at  6$.     At  7$. 

13.  $2345  for  8  mo.  19  da.,  at  6$.     At 

14.  $1846  for  5  mo.  23  da.,  at  6$.     At 

15.  $3456  for  2  mo.  28  da.,  at  6$.     At  4$. 

16.  $5000  for  2  yr.  2  mo.  16  da.,  at  6$.     At  10$. 

In  the  following  examples,  find  the  time  both  by  Compound  Subtraction 
and  exact  days  (3 1C). 

17.  $875  from  May  16,  1888,  to  Jan.  4,  1889,  at  6$.     At 

18.  $412.40  from  Jan.  5,  to  Dec.  12,  at  6$.     At  7£$. 

19.  $1000  from  Mar.  19,  1889,  to  Oct.  5,  1890,  at  6$.    At 

W.  $2420  from  June  17, 1887,  to  Feb.  1,  1888,  at  6$.  At  10$. 
21.  $7000  from  Oct.  12,  1888,  to  May  3,  1890,  at  6$.  At  8$. 
For  additional  examples,  see  Art.  459. 

*  1  year  (363  days)  ax)  days. 


Art.  446.]  INTEREST.  177 


Second  or  6^  Method. 


446.  At  6%,  the  interest  of  one  dollar  for  1  year  is  $0.06. 
For  1  month,  TV  of  a  year,  it  is  TV  of  $0.06,  or  $O.OOJ  (.005). 
For  1  day,  ^  of  a  month,  it  is  ^  of  $0.005,  or  $0.000£. 

Ex.  What  is  the  interest  of  $864,  at  6%,  for  2  yr.  7  mo.  20  da.  ? 

OPERATION.  ANALYSIS.  —  If  the  interest  of  $1 

2  x  .06       =  .12  864         for  1  yr.  is  $.06,  for  2  yr.,  it  is  twice 

7X.OOJ-    =.035  .1581      W-~«r  $.1$  ;-*£  -4fce 

for  1  mo-  is     -°°     for 


90  v   0001.  -  -    OO'H 

*88         times   $.00fc  or  $.035.     If  the  inter- 
6912         est  of  $1  for  1  day  is  $.000|,  for  20 
4320  days,  it  is  20  times  $.000|,  or  $.003^. 

Hence  the  interest  of  $1  for  2  yr.  7  mo. 
20  da.  is  $.  12  +  $.035  +  $.003^,  or 


.  .,        .. 

$136.  800         The  interest  of  $864  is  864  times  $.158$, 
or  $136.80. 

447.  RULE.  —  Take  as  the  interest  of  one  dollar,  six  cents 
for  each  year,  one-half  cent  (or  five  mills)  for  each  month, 
and  one-sixth  of  a  mill  for  each  day.     Multiply  the  prin- 
cipal by  the  sum  of  these  amounts  (as  an  abstract  number). 
The  product  will  be  the  interest  at  6%. 

To  find  the  amount,  add  the  principal  to  the  interest. 

NOTES.  —  1.  In  using  this  method,  to  multiply  by  f  ,  write  ^  twice  ;  to  mul- 
tiply by  f  ,  take  \  and  ^. 

2.  If  the  time  is  less  than  60  days,  and  the  rate  is  6^  or  less,  reckon  the 
interest  on  the  nearest  number  of  dollars.  The  result  will  be  sufficiently 
accurate. 

448.  The  interest  for  any  other  rate  may  be  found  from  the 
interest  at  6%  as  follows  :   At  \%,  divide  by  6  ;  at  1J$,  divide  by 
4  ;  at  2%,  divide  by  3  ;  at  3%,  divide  Tby  2  ;  at  4$,  subtract  \  ;  a'; 
4£%,  subtract  £  ;  at  5%,  subtract  £  ;  at  b\%,  subtract  ^  ;  at  5^, 
subtract  ^  ;  at  6£$,  add  ^  ;  at  6f  %,  add  -J-  ;  at  1%,  add  J-  ;  at 
7|$,  add  J  ;  at  8%,  add  1  ;  at  9$,  add  J  ;  at  10%,  divide  by  6,  and 
multiply  by  10  by  placing  the  point  one  place  to  the  right  ;  at 
12%,  multiply  by  2.     At  any  per  cent.,  divide  by  6  and  multiply 
by  the  rate. 

449.  For  examples  to  be  worked  by  the  above  method,  see 
Art.  445  and  Art.  459- 


178  INTEREST.  [Art.  450. 

Third  or  Month  Method. 

450.  Ex.  What  is  the  interest  of  $875  for  2  yr.  8  mo.  14  da.9 

at  4%  ? 

OPERATION. 

$875 
.04  i 

12  )  35.00       Int.  for  \yr.  ANALYSIS.— The  interest  of  $875  for  1  yr. 

~~7  at  ±%  is  $35  (.04  x  $875),  and  for  1  mo.  is  ^  of 

2.917     Int.  for  l  mo.  ^  or  $2.917.   2  yr.  8  mo.  =  32  mo.    14  da.  = 

32. 4|  Number  of  mos.  .32  m0t    (To  reduce  days  to  tenths  of  a  month, 

0,^2  divide  the  number  of  days  by  3  and  place  the 

Q)^rt  point  one  place  to  the  left.)    If  the  interest  for 
1  mo.  is  $2.917,  for  32.3f  mo.,  it  is  32.3|  times 

11668  $2.917,  or  $94.71. 
5834 

8751 
$94.7052 

451.  RULE. — Multiply  the  principal  by  the  rate  per  cent., 
and  divide  the  product  by  12.     The  result  will  be  the  inter- 
est for  one  month  at  the  given  rate  per  cent.    Multiply  this 
result  by  the  total  number  of  months,  the  years  being  re- 
duced to  months  and  the  days  being  expressed  as  tenths  of  a 
month. 

452.  For  examples  to  be  worked  by  this  method,  see  Art. 
445  and  Art.  459. 

Fourth  or  60-day  Method  at  Q%. 

453.  6%  for  12  months  or  1  year,  is  equivalent  to  \%  for  2 
months  (60  days),  or  -J-  of  one  year.     \%  of  any  amount  is  readily 
ascertained  by  placing  the  point  two  places  to  the  left.     Hence 
the  interest  of  any  sum  at  6$  per  annum  for  2  months,  or  60 
days,  may  be  found  by  placing  the  point  two  places  to  the  left. 
The  interest  for  6  days  may  be  found  by  placing  the  point  three 
places  to  the  left. 

NOTE. — It  will  be  found  advantageous  to  use  a  perpendicular  line  as  a 
separatrix  in  solving  examples  by  this  method.  All  necessity  for  pointing  off 
will  then  be  dispensed  with,  and  confusion  prevented. 


Art.  453.]  INTEREST.  179 

Ex.  1.    What  is  the  interest  of  $1236  for  80  da.,  at  §%  ? 


$12 


OPERATION.  ANALYSIS.—  The  interest  of   $1286  at 

36  =  int.  for  60  da.  Q%  for  60  da.  is  found  to  be  $12.36.  by 
12  =  "  "  20  da.  the  process  already  explained.  If  the 
interest  for  60  da.  is  $12.36,  for  20  da. 


;0  da.         Q  of  60)j  it  win  be  j.  of  |13t365  or 


95 


$16 

Hence  for  80  da.,  it  will  be  $12.36  plus  $4.12,  or  $16.48. 

Ex.  2.  Find  the  interest  of  $864  for  1  yr.  10  mo.  15  da.,  at  §%. 

OPERATION.  ANALYSIS.— The  interest  of 

64  =  int.  for  2  mo.,  or  60  da.         $864  at  6%  for  2  mo.  is  $8.64. 
11  For  1  yr.  10  mo.  (22  mo.),  it 

will    be    11    times    $8.64,    or 
04  =  int.  for  22  mo.  ^  04      If   the  interegt  for 

16  —    '  15  da.  60    da.   is    $8.64,   for  15  da. 

$97    20  =  required  interest.  &  of  60>>  {i  wil1  *»  *  of  $8-64' 

or  $2.16.     Hence  the  interest 

for  the  given  time  will  be  $95.04  plus  $2.16,  or  $97.20. 

Ex.  3.    What  is  the  interest  of  $375.60  for  55  days,  at  §%  ? 

'        „ ..    7  ANALYSIS. — 55  days  =  60  days  Jess 

$3     75.60  =  int.  for  60  da.         5  dayg>     The  interjt  for  6Q  dyays  fa 

_^  "  $3.756,  for  5  days  (TV  of  60),  $.313,  and 


3  |  443      =    "       "    55  "  for  55  davs>  $3.443  ($3.756  -  $.313). 

454.  Aliquot   Parts   of  60.— 1  =  ¥V  5   2  =  ^ ;  3  =  ^  ; 
4  =  ^;    5  =  TV;    6=.TV;    10  =  i;  12  =  i  ;  15  =  i;  20  =  i  ; 
30  =  i. 

NOTES. — 1.  To  divide  by  10,  place  the  figures  of  the  basis  one  place  to 
the  right. 

2.  To  divide  by  20,  30,  or  60,  divide  by  the  first  figure  and  write  the 
quotient  figures  one  place  to  the  right. 

455.  Ex.  1.  What  is  the  interest  of  $976  for  26  days,  at  6^? 

OPERATION  ANALYSIS. — As  26  is  not  an  aliquot 

76     =  int.  for  60  da.        part  (388)  of  60,  take  20,  which  is  $  of 
pj-q  __    ((       (f    ^7>    7  60,  and  6,  which  is  -^  of  60.     Divide  the 

basis  which  is  the  interest  for  60  da.  by  3 

9?6  —  '     _?  cla'         to  find  the  interest  for  20  da.  ($3.253) ;  and 

229  =    ({       "     26  da.         the  same  sum  by  10,  to  find  the  interest 

for  6  da.   ($.976).      (See  Art.  454,  1.) 
The  sum  of  these  two  results  will  be  the  interest  for  26  days. 


180  INTEREST.  [Art.  455. 

Ex.  2.  Find  the  interest  of  $732.80  for  2  yr.  8  mo.  27  da., 
at  6%  ? 

OPERATION.  ANALYSIS.— The  interest 

32.8  =  int.  for  2  mo.,  or  60  da.       for  2  mo.,  forming  the  basis, 
16  is  $7.328.     Multiply  this  by 

——  16  to  find  the  interest  for  32 

mo.  (2  yr.  8  mo.}.    27  =  15  + 
12.     To  find  the  interest  for 

832   =  int.  for  15  da.  15  da.,  divide  the  basis  by  4 

466   =    "       "    12  da.  (15  =  £  of  60)  ;  and  the  same 

sum  by  5  to  find  the  interest 
546  -.=  required  int.  for  ia  da<    By  adding  these 

results,  we  have  the  interest  for  the  given  time  at  Qfc. 

456.  If  the  number  of  days  given  is  not  an  aliquot  part  of 
60,  it  will  need  to  be  so  separated  that  the  component  parts  will  be 
aliquot  parts  of  60. 


43 

73 

1 

1 


Numbers  not  aliquot  parts  of  60,  with  best  divisions  :  7  =  6  +  1;  8  =  6  +  2; 
9  =  6  +  3;  11=6  +  5,  or  10  +  1;  13=10  +  3;  14  =  12  +  2;  16  =  10  +  6;  17  = 
12  +  5,  or  15  +  2;  18  =  12  +  6.  (The  interest  for  18  days  may  be  found  by 
multiplying  the  basis  by  3,  and  placing  the  figures  of  the  product  one  place 
to  the  right);  19  =  15  +  4,  or  10  +  6  +  3;  21  =  15  +  6;  22  =  20  +  2(2  =  TV of  20); 
23  =  20  +  3 ;  24  =  12  + 12  (or  multiply  by  4  and  place  the  figures  of  the  product 
one  place  to  the  right);  25  =  20  +  5  (5  =  |  of  20);  26  =  20  +  6  ;  27  =  15  +  12  ; 
28  =  12  +  12  +  4  (4  =  i  of  12),  or  20  +  6  +  2;  29  =  12  +  12  +  5,  or  20  +  6  +  3. 

457.  RULE. — Draw  a  perpendicular  line  two  places  to  the 
left  of  the  decimal  point ;  the  result  will  be  the  interest  at  6% 
for  2  months,  or  60  days,  the  dollars  being  on  the  left,  and 
the  cents  on  the  right  of  this  line.     Multiply  this  result  by 
one-half  the  total  number  of  months.     To  this  product,  add 
that  proportion  of  the  interest  for  60  days,  which  the  given 
number  of  days  is  of  60. 

XOTE. — The  interest  at  any  other  rate  per  cent,  may  be  found  as  in 
Art.  448. 

458.  The  interest  at  6%  may  be  found  for  6  days  by  placing 
the  point  three  places  to  the  left  (453).    In  many  examples,  when 
the  time  is  less  than  100  days,  the  process  is  shortened  by  taking 
as  the  basis  the  interest  for  6  days  instead  of  60  days. 

Ex.  Find  the  interest  of  $375  for  8  days  at  6#.  $425  for  79 
days.  $500  for  47  days. 


Art.  458.] 

1ST  OPERATION. 

375     6  da. 
125     2  da. 
8  da. 


500 


INTER  E  ST. 


$ 

2ND  OPERATION. 

425       6  da. 
525     78  da.  (13x6) 
071       1  da. 

•i 

$ 

ED  OPERATION. 

500       6  da. 

5 

5 

4 

000 

083 

48  da. 
1  da. 

596     79  da. 

3 

917 

47  da. 

EXAM  PLES. 


459.  Find  the  interest  of  the  following  at  6$.     (See  Arts. 
453  and  454.) 


1.     $864  for  60 
$396  for  20 

$290  for  72 
$785  for  66 
$636  for  62 
$400  for  90 
$525  for  61 
$600  for  10 
$728  for  65 
$340  for  15 


2. 
3. 

4. 

5. 
6. 
7. 
8. 
9. 
10. 


days, 
days, 
days, 
days, 
days, 
days, 
days, 
days, 
days, 
days. 


11. 
12. 
13. 

14- 
15. 

16. 
17. 
18. 
19. 

20. 


What  is  the  interest  of 

r 

21.  $375.60  for  8  mo.  20  da.,  at 
rt.  453.) 

22.  $1727  for  7  mo.  -tfrfla.,  at  ^  ?    At 

23.  $449.38  for  1  yr.  4  mo.  12  da.,  at 

24.  $285  for  1  yr.  5  mo.  10  da.,  at 

25.  $432.65  for  2  yr.  2  mo.  6  da., 

26.  $1235  for  2  yr.  5  mo.  5  da.,  at'M  ? 

#7.  $445.25  for  5  mo.  4  o^.,  at  6^:?     At  9$ 
0£.  $2440.50  for  97  days,  at  6/?     At  1% '? 
455.) 

#0.  $3125  for  38  days,  at  6; 
$247.50  for  69  days,  at 
$512.45  for  5  mo.   11 
Art.  455.) 

$1478  for  1  yr.  2  mo.  13  da.,  at 
$2810.60  for  9  mo.  24  da.,  at  6^ 
$944.50  for  1  yr.  10  mo.  22  da., 
$575  for  2  yr.  8  mo.  \16  da.,  at  ( 


$1275  for  50  days. 
$2345  for  30  days. 
$1728  for  63  days. 
$375.60  for  5  days. 
$414.80  for  54  days. 
$1024  for  59  days. 
$2375  for  90  days. 
$1000  for  57  days 
$2480  for  63 
$5000  for 


?    At  5^  ?     (See  Ex.  2, 


#?    (448) 

?    At  7$  ? 
At  5$? 
?    At  8^? 

At  4^? 


(See  Ex.  1,  Art. 


?    At  7 

%  ?    At 

.    at  Q 


? 


?    At  7^  ?     (See  Ex.  2, 


? 
at 


?     At  8 
At  6f0  ? 


? 


? 
At 


At 


INTEREST. 


[Art.  459. 


u/ 


36.  $1112  for  3  mo.  14  da.,  at  6%  ? 
$5285  for  1  yr.  6  mo.  21  da.,  at 


38.  $7218  for  11  mo.  18 
Find  the  amount  of 


at  6^  ? 


At 
%  ? 
At 


At 


At 


At  8< 

At' 


$416.75  for  8  mo.  17  da.,  at  6% 
$1235  for  2  yr.  1  mo.  19  da.,  at  6^ 

41.  $575.60  for  1  yr.  4  mo.  23  da.,  at  6< 

42.  $2214  for  4  mo.  25  da.,  at  6%  ?    At 
4#.  $6315  for  5  mo.  29  da.,  at  6%  ?     At 
4£  $4312  for  4  mo.  26  da.,  at  6%  ?    At 
45.  $384.30  for  2  mo.  28  da.,  at  Q%  ?    At 
40.  $1296  for  1  yr.  11  mo.  27  da.,  at  6%  ? 
47.  $4375  for  2  yr.  8  mo.  24  da.,  at  Q%  ? 

Find  the  interest  of  the  following  at  §%.     (See  Art.  458.) 


At 
At 


48.  $2000  for  6  dc 

49.  $1728  for  37  dc 

50.  $3485  for  92  dc 

51.  $1234  for  69  dc 

52.  $375.60  for  8 


63.  $3748  for  53  da.  (54-1.) 

64.  $4126  for  89  da. 
66.  $1289  for  39  da. 
66.  $4000  for  17  da. 
£7.  $2000  for  28  da. 


NOTE.  —  Find  the  time  in  the  following  examples  both  in  months  and  days, 
and  in  exact  days  (3  1C). 


J&1234  from  May  10  to  Dec.  4,  at  5^  ?    At 
69.  $444.40  from  Jan.  13  to  Nov.  2,  at  4%  ?    At 

1575.20  from   June   5,   1882,   to   Feb.   4,   1883, 


at 


0.7.  $2375   from   July   17,   1884,  to   Nov.   27,   1885,   at 
(Exact  time,  1  yr.  133  da.) 

$3212   from   Aug.   24,   1881,   to   Jan.   20,   1884,  at 

? 
63.  $475.80  from   May   12,   1882,   to  Feb.   1,   1884,    at 


At 


At 


64.  Find  the  interest  of  $180  for  253  days,  at  Q 

NOTE.  —  In  many  examples,  labor  can  be  saved  by  having  the  time  and 
principal  exchange  places.  In  the  above  example,  the  interest  of  $180  for 
253  days  is  the  same  as  $253  for  180  days  ($2.53  x  3). 

66.  Find  the  interest  of  $600  for  173  days  at  9#.     At  4#. 
66.  Find  the  interest  of  $3000  for  111  days  at  12$.     At  3#. 


Art.  459.]  INTEREST.  183 

Find  the  interest  of 

67.  $1800  from  Jan.  17  to  Oct.  2,  at  6#.  At 
££.  $540  from  May  11  to  Dec.  18,  at  5%.  At 
£2.  $3000  from  Feb.  4  to  July  13,  at  4$.  At 
70.  $2400  from  July  13  to  Dec.  1,  at  5$.  At 
7.7.  $600  from  Aug.  16  to  Nov.  24,  at  6$.  At 
72.  $1200  from  May  19  to  July  3,  at  1%.  At 
75.  $480  from  March  13  to  Sept.  3,  at  Sfc.  At 

74.  $720  from  Feb.  27  to  May  15,  at  9$.     At 

75.  $2100  from  Sept.  2  to  Nov.  30,  at  10$.     At 

76.  If  $9200  is  loaned  Sept.   18,   1882,  at  6$,  what  is  due 
May  9,  1885  ?     (Time  by  C.  S.) 

77.  What  is  a  banker's  gain  in  1  year  on  $10000  deposited  at 
6$,  and  loaned  11  times  at  \\%  a  month  ? 

78.  A  note  for  $1421,  with  interest  after  4  months,  at  7$,  was 
given  Dec.   1,   1881,   and   paid  Aug.   12,   1883.     What  was  the 
amount  due  ?     (C.  S.) 

79.  Nov.  6,  1881,  I  bought  a  lot  of  grain  for  $753.20  ;  Dec.  1G, 
I  sold  a  part  of  it  for  $375.60  ;  and,  Dec.  31,  I  sold  the  remainder 
for  $411.40.     Money  being  worth  6%,  how  much  did  I  gain  by  the 
transaction  ? 

80.  A  merchant  marks  his  goods  with  two  prices,  the  one  for 
cash  and  the  other  for  4  months'  credit.     If  the  cash  price-  is  $28, 
what  ought  the  credit  price  to  be,  money  being  worth  10$  ? 

81.  May  27,  $328  is  loaned  at  6$,  and  Aug.  16,  $1000  is  loaned 
at  5J-%.     What  is  the  total  amount  due  Dec.  11  ? 

82.  A  banker  borrows  $100000  at  3J$,  and  pays  the  interest 
at  the  end  of  the  year  ;  he  loans  it  at  5$  and  receives  the  interest 
semi-annually.     How  much  does  he  gain  in  one  year,  if  he  loans 
the  semi-annual  interest  until  the  end  of  the  year  ? 

83.  A  buys  a  bill  of  goods  amounting  to  $2776.40,  on  the  fol- 
lowing terms: — "4  months,  or  less  5$  cash."     He  accepts  the 
latter  terms,  and  borrows  the  money  at  6$  to  pay  the  bill.     How 
much  does  he  gain  ? 

84-  A  person  buying  a  building  lot  for  $5400,  agreed  to  pay  for 
it  in  four  equal  semi-annual  installments,  with  interest  at  6% ; 
what  was  the  total  amount  of  money  paid,  the  first  payment 
being  made  at  the  time  of  the  purchase  ? 

85.  A  banker  borrows  $10000  at  4J$,  and  lends  half  of  it  at  6$ 
and  half  at  8%.  What  does  he  gain  in  2  yr.  4  mo.  26  da.  ? 


184  INTEREST.  [Art.  46O. 


ACCURATE     INTEREST. 

46O.  To  find  the   accurate  interest  (365  days  to  the 
year)  for  any  rate  and  time.     (See  Art.  437.) 

Ex.    What    is   the   accurate   interest   of    $865,    at   4$,    from 
June  21  to  Dec.  13  ? 

OPERATION. 

$865     Principal.  ANALYSIS.  —  From  June  21  to  Dec.  13, 

there  are  175  days.     The  interest  of  $865 
for  1  yr.,  at  4%,  is  $34.60.     For  175  days, 


34.60      Interest  for  1  yr. 
m  m  of  1  yr.,  it  is  HI  of  $34.60  -- 

°r  $1°'59- 


365  )  6055.00  (  16.59 

461.  KULE. — Multiply  the  principal  by  the  rate  per  cent, 
expressed  decimally.  The  result  will  be  the  interest  for 
one  year. 

Multiply  the  interest  for  one  year  by  the  number  of  days, 
and  divide  the  product  by  365. 

NOTES. — 1.  When  the  number  of  days  is  a  multiple  of  5,  multiply  by  £ 
the  number  of  days,  and  divide  the  product  by  73.  In  the  above  example, 
$865  x  .04  x  35  -f-  73  =  $16.59. 

2.  To  find  the  interest  at  any  per  cent.,  multiply  by  twice  the  rate  as  an 
integer,  by  the  number  of  days,  divide  the  product  by  73,  and  point  off  3 
places.    In  the  above  example,  $865  x  8  x  175  -*-  73000  =  $16.59. 

3.  To  find  the  interest  at  5  % ,  multiply  tbe  principal  by  the  number  of 
days,  divide  the  product  by  73,  and  point  off  2  places.     From  this  result  to 
find  the  interest  at  Qft>,  add  \  ;  4^%,  subtract  •£$  ;  4^,  subtract  \. 


462.  Accurate  Interest  from  Ordinary  Interest.  —  The 

difference  between  ordinary  interest  and  accurate  interest  for 
1  day  equals  the  difference  between  -^fa  and  ^T  of  a  year's 
interest. 

1  1      _  365  —  360  _  5  J5_        _1_       J^ 

360  ~~  365  ~  365  x  360  ~~  365~x~360  ~~  365  °     360  ~  73  ° 
1  5  J>          _1  l_     .  J_ 

~~  ~~         °  * 


360   3651T366  ~~  360    365  ~~  72    365 

The  difference  between  the  two  methods  is  ^  of  ordinary  interest,  or  -j-% 
of  accurate  interest  (437,  Note  4).  Therefore,  from  ordinary  interest  to  find 
accurate  interest  subtract  y1^. 


Art.  462.]  ACCURATE    INTEREST.  185  ' 

In  reckoning  accurate  interest,  on  account  of  the  many  short  methods  of 
ordinary  interest,  many  accountants  prefer  to  calculate  ordinary  interest  first, 
and  then  make  the  necessary  deduction. 

Since  -fa  is  about  \\%,  the  following  approximate  method  may  be  used 
in  reducing  ordinary  interest  to  accurate  interest  :  From  the  ordinary  interest 
subtract  1%  and  \%  of  itself. 

Ex.     Eeduce  $32.70  ordinary  interest  to  accurate  interest. 

NOTE.—  The  exact  result  should  be  $32.252.  The 
results  by  this  method  are  too  great  by  1  cent  for  each  $27 
interest  ;  $.036  for  each  $100  interest  ;  $.36  for  each  $1000 
interest.  Where  greater  accuracy  is  required,  the  neces- 
sary  correction  can  be  made. 

EXAMPLES. 

463.  What  is  the  accurate  interest  of 

L  $43£.32,  at  6#,  for  25  days  ?  5.  $292,  at  $±%,  for  140  days  ? 

$6030,  at  5$,  for  141  days?    6.  $438,  at  6$,  for  210  days? 

8.  $780,  at  6#,  for  90  days?       7.  $350,  at  4$,  for  150  days? 
4.  $437.80,  at  1%,  for  63  days?  8.  $500,  at  4J&  f°r  100  days? 

9.  $3110.45,  at  5J^,  for  90  days? 

$$73.70,  at  7$,  from  June  4  to  Dec.  28  ? 

11.  $500,  at  Q%,  from  July  24,  to  Sept.  16  ? 

12.  $365,  at  5%,  from  June  30  to  Dec.  21  ? 

13.  $1080,  at  Q£  from  May  9,  1878,  to  Jan.  30,  1879  ? 

14.  $1728,  at  7#,  from  Jan.  6,  1878,  to  Jan.  21,  1880  ? 

N  ./J.  Eequired  the  exact  interest  on  three  U.  S  bopds  of  $5000 
lr 


eaclr  at  3J^,  from  July  1  to  Aug.  11. 

16.  What  is  the  interest  on  three  U.  S.  bonds  of  $1000  each,  at 
4J^,  from  Sept.  1  to  Nov.  15  ?  .^ 

'-^ZL  What  is  the  interest  on  a  $5000  U.  S.  bond,  at  4$,  from 
Oct.  1  to  Dec.  16  ? 

18.  What  is  the  interest  on  a  U.  S.  bond  of  $1000,  bearing 
Z\%  interest,  from  May  1  to  July  19  ? 

19.  What  is  the  interest  on  a  $500  U.  S.  bond,  at  4$,  from 
Apr.  1  to  May  10.?  f 

20.  What  is  Aae  interest  on  a  $5000  U.  S.  bond  from  Nov.  1, 
1881,  to  Jan.  3,  TJf%,  %f^? 

21.  What  is   the   difference  between  ordinary   and   accurate 
interest  of  $10000  for  219  days  at 


186 


INTEREST. 


[Art.  464. 


PROBLEMS     IN     INTEREST. 

464.  To  find  the  rate,  the  principal,  interest  or  amount, 
and  time  being  given. 

Ex.    At  what  rate  will  $720,  in  1  yr.  4  mo.  10  da.,  produce 
$44. 10  interest? 

OPERATION.  ANALYSIS.  —  The    interest    on    a 

given  principal  for  a  given  time  is  in 
proportion  to  the  rate  per  cent.  At  one 
per  cent.  $720  will,  in  1  yr.  4  mo.  10  da., 
produce  $9.80  interest.  To  produce 
$44.10  interest,  the  required  rate  must 
be  as  many  times  \%,  as  $9.80  is  con- 
tained times  in  $44.10,  or  4£  times. 


$7 

57 

_1 

6  )58 


20 

_8 
60 
20 

80 


80  )  $44. 10  (  4J  Ans.       Hence  the  answer  is 


465.  RULE.  —  Divide  the  given  interest  by  the  interest  of 
the  given  principal,  for  the  given  time,  at  1%. 

NOTE.  —  When  the  amount  is  given,  find  the  interest  by  subtracting  the 
principal  from  the  amount. 


466.  At  what  rate  will 
f  1.  $864  in  8  mo.  10  da.  produce  $42  interest  ? 

2.  $1000  in  9  mo.  9  da.  produce  $54.25  interest  ? 

8.  $852  in  1  yr.  7  mo.  16  da.  amount  to  $935.21  ? 

4.  $1926  in  2  yr.  8  mo.  24  da.  produce  $263.22  interest  ? 
•v.  5.  $375.60  in  1  yr.  10  mo.  22  da.  amount  to  $425.41  ? 
^6.  $1872  in  7  mo.  17  da.  produce  $41.31  interest  ? 

7.  $435.60  in  1  yr.  2  mo.  18  da.  amount  to  $478  ? 

8.  $1338.72  in  6  mo.  27  da.  produce  $34.64  interest  ? 

9.  $1728  in  8  mo.  21  da.  amount  to  $1778.11  ? 

10.  $3456  in  5  mo.  8  da.  produce  $91.01  interest  ? 

11.  $5280  in  11  mo.  11  da.  amount  to  $5720.12  ? 

12.  $1234  in  8  mo.  22  da.  produce  $80.83  interest  ? 

13.  $6975  in  3  mo.  28  da.  amount  to  $7215.06  ? 

14.  $525  in  1  yr.  11  mo.  18  da.  produce  $309.75  interest? 

15.  $500  in  3  yr.  11  mo.  12  da.  amount  to  $658  ? 

16.  $4680  in  2  yr.  6  mo.  11  da.  produce  $710.58  interest  ? 

17.  $614.45  in  162  days  amount  to  $633.805  ? 


Art.467.] 


PROBLEMS     IN    INTEREST. 


187 


467.To  find  the  time,  the  principal,  interest  or  amount, 
and  rate  being  given. 

Ex.    In  what  time  will  $426,  at  Q%,  produce  $59.427  interest  ? 


OPERATIONS. 


$426 
.06 

5^56  )  $59.427  (  yr.  2.325 
51  12       12 
~8  307  mo.  3.900 
7  668       30 
6390  da.  ^7.000 
5112 


Or,   $426 


$25.56  )  $59.427  (  2  yr. 
51  12 

8.307 
12 


$25.56  )  99. 684  (3  mo. 
76.68 
23.004 
J30 

$25.56  )  690. 120  (M  da. 

ANALYSIS. — The  interest  on  a  given  principal  at  a  given  rate  %  is  in 
proportion  to  the  time.  In  one  year  $426,  at  6fc,  will  produce  $25.56 
interest.  To  produce  $59.427  interest,  it  will  require  as  many  years  as  $25.56 
is  contained  timefe  in  $59.427,  or  2.325  yr.  2.325  yr.  equal  2  yr.  3  mo.  27  da. 
(289). 

468.  RULE. — Divide  tJ^e  gimn^interest  by  the-  interest  of 
the  given  principal,  at  the  given  rate,  for  one  year. 

The  integral  part  of  the  qwotient  will  be  years.  Reduce 
the  decimal,  if  any,  to  months  and  days  (289). 


EXAMPLES. 

469.  In  what  time  will 

1.  $3000,  at  7$,  produce  $108.50  interest  ? 

2.  $1728,  at  6%,  amount  to  $1872  ? 

8.  $3932,  at  7%,  produce  $597.88  interest  ? 

4.  $735,  at  5%,  amount  to  $742.66  ? 

5.  $1222.25,  at  §%,  produce  $39.52  interest  ? 

6.  $375.60,  at  7#,  amount  to  $425.41  ? 

7.  $1461.75,  at  Q%,  produce  $420.25  interest  ? 

8.  $1200,  at  3f  %,  amount  to  $1413  ? 

9.  $4500,  at  5%,  produce  $181.25  interest  ? 
10.  $276.50,  at  10$,  amount  to  $303.46  ? 


188  INTEREST.  [Art.  469. 

In  what  time  will 

11.  $1020,  at  6$,  produce  $89.25  interest  ? 

12.  $6495,  at  1%,  amount  to  $7161.81  ? 
IS.  $100,  at  6%,  produce  $100  interest  ? 

14.  $125,  at  7$,  amount  to  $375  ? 

47O.  To  find  the  principal,  the  interest,  time,  and  rate 
being  given. 

Ex.    What  principal  will  produce  $152.64  interest,  in  1  yr. 
5  mo.  20  da.,  at  §%  ? 

OPERATION. 

$.088^)  $152.64  (1728 
_3  __  3_ 

.265    )    457.920  ANALYSIS.  —  The  interest  on  any  principal 

265  is  as  many  times  greater  than  the  interest  of 

$1,  as  that  principal  is  greater  than  $1.     One 
doUar,  in  1  yr.  5  mo.  20  da.,  at  6%  (447),  will 
1855  produce  $.088^  interest.  To  produce  $152.64,  the 

742  principal  must  be  as  many  times  $1  as  $.088£  is 

contained  times  in  $152.64,  or  $1728. 


2120 
2120 

471.  RULE.  —  Divide  the  given  interest  ~by  the  interest  of 
$1  for  the  given  time,  at  the  given  rate. 

EXAM  PLES. 

472.  What  principal  will  produce 

1.  $1235  interest,  in  1  yr.  8  mo.  12  da.,  at  Q%  ? 

2.  $49.81,  in  9  mo.  24  da.,  at  1%  ? 

3.  $186.75,  in  1  yr.  4  mo.  20  da.,  at  6%  ? 

4.  $244.44,  in  7  mo.  18  da.,  at  5%  ? 

5.  $375.60,  in  2  yr.  4  mo.  6  da.,  at  8%  ? 

6.  $54.25,  in  3  mo.  3  da.,  at  1%  ? 

7.  $387.40,  in  2  yr.  8  mo.,  at  ±\%  ? 
£.  $456,  in  93  da.,  at  6%  ? 

9.  $375,  in  63  da.,  at  1%  ? 
10.  $1000,  in  1  yr.  18  d«.,  at  ?>%  ? 
Ji   $538.80,  in  10  mo.  24  da.,  at  5%  ? 
12.  $416.75,  in  8  mo.  21  da.,  at  4%  ? 
7#.  $645.39,  in  4  yr.  8  mo.  10  da.,  at  4$  ? 


Art.  473.]  PROBLEMS.  189 

473.  To  find  the  principal,  the  amount,  time,  and  rate, 
being  given. 

Ex.     What  principal  will  amount  to  $1880.64,  in  1  yr.  5  mo, 
20  da.,  at  §%  ? 

OPEBATION. 

$1.088$)  $1880. 64  (1728. 

3.265    )    5641.920  ANALYSIS. -The  amounts  of  different 

principals  for  the  same  time  and  rate  % ,  are 
23769  to  each  other  as  the  principals.     One  dollar, 

22855  in  *  yr-  5  mo-  ^O  da.,  at  6%  will  amount  to 

$1.088$.     To  amount  to  $1880.64,  the  prin- 
cipal must  be  as  many  times  $1  as  $1.088$ 
6530  are  contained  times  in  $1880.64,  or  $1728. 

26120 

26120 

0 

474.  RULE. — Divide  the  given  amount  by  the  amount  of 
$1  for  the  given  time,  at  the  given  rate. 

EXAMPLES. 

475.  What  principal  will  amount  to 
1.  $1272.254,  in  6  mo.  6  da.,  at  6%? 
&  $5538.72,  in  8  mo.  12  da.,  at  7%  ? 

8.  $3695.04,  in  1  yr.  4  mo.  18  da.,  at  5^? 

4.  $442.71,  in  2  yr.  2  mo.  24  da.,  at  8$? 

5.  $14794.31,  in  3  yr.  3  mo.  3  da.,  at  6%  ? 

6.  $1793.38,  in  7  mo.  17  da.,  at  6$? 

7.  $1010.65,  in  5  yr.  8  mo.  6  da.,  at  7%  ? 
A  $977.75,  in  1  yr.  10  mo.  10  da.,  at 

9.  $1716.75,  in  3  yr.  4  mo.  21  da.,  at  4 
.70.  $2808.08,  in  2  yr.  8  mo.  12  da,,  at 

11.  $4312.22,  in  1  yr.  2  mo.  11  da.,  at 

12.  $6528.49,  in  4  yr.  7  mo.  6  da.,  at 

13.  $1763.02,  in  1  yr.  2  mo.  21  da.,  at 

14.  $2457.28,  in  2  yr.  5  mo.  23  da.,  at 

15.  $5375.34,  in  1  yr.  6  mo.  15  da.,  a 

16.  $3536.87,  in  2  yr.  7  mo.  10  da.,  at  9 

17.  $4221.50,  in  3  yr,  10  mo.  27  da.,  at 


190  INTEREST.  [Art.  476. 


PRESENT  WORTH  AND  TRUE  DISCOUNT. 

476.  The  Present  Worth  of  a  debt  due  at  some  future  time 
is  its  value  now.     Theoretically,   it  is  a   sum  that,  if  placed  at 
interest  to-day  for  the  given  time,  would  amount  to  the  face  of 
the  debt. 

477.  The  True  Discount  is  the  diiference  between  the  face 
of  the  debt  and  the  present  worth. 

This  subject  is  an  application  of  the  principle  illustrated  in  Art.  473,  the 
face  of  the  debt  being  the  amount,  the  present  worth  the  principal,  and  the 
true  discount  the  interest. 

In  actual  business  true  discount  is  little  used,  banks  and  merchants 
generally  using  bank  discount  (496).  True  discount  is  the  interest  on  the 
present  worth  for  the  given  time,  while  bank  discount  is  interest  on  the  face 
of  the  debt.  The  difference  is  therefore  equivalent  to  the  interest  on  the  true 
discount.  For  discount  on  bills,  etc.,  when  time  does  not  enter  as  an 
element,  see  Art.  415. 

Ex.  Mr.  B  owes  me  $212,  payable  one  year  from  to-day  with- 
out interest ;  what  is  the  present  worth  of  the  debt,  the  current 
rate  of  interest  being 


ANALYSIS. — Since  $1  in  one  year,  at  Q%,  amounts  to  $1.06,  it  would 
require  as  many  dollars  to  amount  to  $212,  as  $1.06  is  contained  times  in 
$212,  or  $200.  The  true  discount  is  $212- $200,  or  $12. 

478.  EULE. — /.    To  find  the  present  worth,  divide  the 
face  of  the  debt  by  the  amount  of  $1  for  the  given  time, 
at  the  given  rate. 

II.  To  find  the  true  discount,  subtract  the  present  worth 
from  the  face  of  the  debt. 

EXAMPLES. 

479.  The   current  rate  of  interest  being   6$,  what  is  the 
present  worth  and  true  discount  of 

1.  $1000,  due  2  years  hence  ?      8.  $600,  due  in  1  yr.  7  mo.  ? 

.    2.  $500,  due  in  2  yr.  4  mo.  ?        4.  $800,  due  in  9  mo.  24  da.  ? 

5.  $325,  due  in  2  yr.  5  mo.  12  da.  ? 

6.  $175,  due  in  1  yr.  4  mo.  16  da.  ? 

7.  $800,  due  in  5  yr.  8  mo.  22  da.  ? 

8.  $900,  due  in  6  yr.  8  mo.  14  da.  ? 


Art.  479.]  REVIEW    EXAMPLES.  191 

9.  Mr.  C.  desiring  to  pay  a  bill  of  $1728  4  months  before  it 
was  due,  was  allowed  a  discount  equivalent  to  the  interest  on  the 
face  of  the  bill  for  the  unexpired  time  at  6^  per  annum  (bank 
discount).     How  much  greater  was  this  discount  than  the  true 
discount  ? 

10.  Goods  to   the   amount   of   $3750  are  sold  on  a  credit  of 
4  months.     For  how  much  cash  could  the  merchant  afford  to  sell 
the  same  goods,  money  being  worth  10%  per  annum  ? 

11.  If  $10000  will  be  due  me  May  28,  and  $8000  May  16,  what 
discount  should  I  make  on  the  two  claims  Apr.  1,  money  being 
worth  S%? 

REVIEW     EXAMPLES, 

48O.  1.  What  is  the  interest  of  $375.60,  for  1  yr.  10  mo.  16 
da.,  at  Q%  ? 

2.  What  is  the  amount  of  $1765  for  7  mo.  20  da.,  at  7^? 

3.  At  what  rate  will  $1234,  in  2  yr.  2  mo.  26  da.,  produce 
$138. 14  interest  ? 

4.  In  what  time  will  $585,  at  Q%,  produce  $67.08  interest  ? 

5.  What  principal  will,  in  1  yr.  8  mo.  14  da.,  at  6%,  produce 
$176.22  interest  ? 

6.  The  semi-annual  interest  on  a  mortgage   at   1%  is   $350. 
What  is  the  face  of  the  mortgage  ? 

7.  Mr.  B.  invests  $49500  in  a  business  that  pays  him  $594  per 
month.     What  annual  rate  of  interest  does  he  receive  ? 

8.  Which  is  the  better  investment,  and  what  per  cent.,  one  of 
$8400,  yielding  $336  semi-annually,  or  one  of  $15000,  producing 
$1425  annually  ? 

9.  May  18th,  a  speculator  bought  1600  bushels  of  wheat,  at 
$1.50  a  bushel.     He  afterward  sold  the  whole  for  $2472  cash,  his' 
profit  being  equivalent  to  8%  per  annum  on  the  amount  invested. 
What  was  the  date  of  the  sale  ? 

10.  The  par  value  of  Mr.  A/s  bank  stock  is  $9000,  and  he 
receives  a  semi-annual  dividend  of  $315.     What  per  cent,  is  the 
dividend  per  annum  ? 

11.  Mrs.  C/s  son  is  now  16  yr.   old  ;    how  much  must  she 
invest  for  him  at  §%,  that,  on  arriving  at  age,  he  may  have,  with 
simple  interest,  $25000  ? 

12.  A  bill  of  goods  amounting  to  $4316.75  is  due  May  27;  how- 
much  would  settle  it  May  1  at  Q%?    How  much  July  3  ? 


192  INTEREST.  [Art.  480, 

18.  A  gentleman  loaned  115000,  at  6$.  Jan.  1, 1880,  interest 
and  principal  together  equalled  $20000.  When  was  the  money 
loaned  ? 

14.  Find  the  interest  on  $3000,  from  Mar.  16  to  Dec.  4,  at 
6%,   by  the  following  methods  (437):  1,   ordinary  interest  and 
compound  subtraction  ;  2,  ordinary  interest  and  exact  number  of 
days  ;  3,  accurate  interest. 

15.  A  man  loaned  another  a  sum  of  money,  payable  in  5 
months,  with  interest  at  the  rate  of  6$,.  and  at  the  end  of  that 
time  received  $666.25  in  return.     How  much  did  he  loan  ? 

16.  A  speculator  borrowed  $10925  at  6$,  May  16,  1882,  with 
which  he  purchased  flour  at  $6.25  per  barrel.     June  11,  1883,  he 
sold  the  flour  at  $7.50  per  barrel,  cash.     What  did  he  gain  ? 

17.  B  bought  225  A.  24  sq.  rd.  of  land,  Aug.  18,  1882,  at  $4 
an  acre,  borrowing  the  money  to  pay  for  it,  at  5%.     He  sold  the 
land  April  7,  1886,  at  an  advance  of  $299.40  on  cost.     If  mean- 
while he  paid  $46.50  for  taxes  on  the  land,  did  he  gain  or  lose, 
and  how  much  ? 

18.  A  speculator  bought  9000  bu.  grain  at  $1.80  per  bushel, 
Mar.  18,  1875,  the  money  paid  for  it  being  borrowed  at  5}%. 
Dec.  12,  1875,  he  sold  f  of  the  grain  at  $2.00  per  bushel,  and  the 
remainder  at  $1.90  per  bushel.     What  was  gained  or  lost  by  the 
transaction  ? 

19.  A  owes  B  £260  9s.  Qd.,  with  interest  at  5%,  for  143  days. 
He  pays  25%  of  the  amount  due  ;  how  much  remains  ? 

NOTE. — In  England,  interest  is  usually  computed  on  the  basis  of  365  days 
to  the  year,  when  the  time  is  given  in  days.  The  legal  rate  in  England  is  5%. 
To  calculate  interest  on  English  money,  reduce  the  shillings  and  pence  to  the 
decimal  of  a  pound  (see  Art.  342,  Ex.  12,  Note),  apply  any  of  the  methods 
under  Art.  461,  and  reduce  the  resulting  decimal  to  shillings  and  pence. 

Find  the  accurate  interest  of 

20.  £425,  from  Aug.  4  to  Dec.  28,  at  5%. 

21.  £625  12s.,  from  Jan.  12  to  Apr.  1,  at  4%. 

22.  £717  16s.  Wd.,  from  Mar.  3  to  June  16,  at  ±\%* 

23.  £429  10s.  Sd.,  from  Sept.  16  to  Nov.  30,  at  3%. 

24.  £516  18s.  3d.,  from  Aug.  1  to  Oct.  18,  at  3J#. 
26.  £612  6s.  lid.,  from  July  1  to  Nov.  3,  at  5%. 

*  When  the  time  is  less  than  1  year,  and  the  rate  is  G%  or  less,  reject  the  pence,  if  less  than 
6 ;  add  1  shilling,  if  more  than  6.  The  result  will  be  sufficiently  accurate. 


Art.  481.]  COMPOUND     INTEREST.  193 


COMPOUND    INTEREST.* 

481.  Compound  Interest  is  interest  not  only  on  the  prin- 
cipal, but  also  on  the  interest  after  it  becomes  due  (4:33). 

1.  Interest  may  be  compounded  annually,  semi-annually,  quarterly,  etc. 

2.  Interest  upon  interest  due,  or  compound  interest,  cannot  be  collected 
•  by  law,  that  is,  payment  cannot  be  enforced ;  but  such  a  payment  is  equitable, 
'  and  the  receiving  of  it,  if  the  debtor  is  willing  or  can  be  induced  to  pay  it,  does 

not  constitute  usury  in  the  legal  sense  of  the  word.  In  the  State  of  Missouri, 
parties  may  contract  in  writing  for  the  payment  of  interest  upon  interest,  but 
it  shall  not  be  compounded  oftener  than  once  a  year. 

Ex.     What  is  the  compound  interest  of  $1000  for  3  years, 

at  6^? 

OPERATIONS. 

$1000.00     Principal.  Or 

60.00     Interest  for  1  yr. 

1060          Amount  for  1  yr.,  or  2d  principal. 
63.60     Interest  of  $1060  for  1  yr. 

1123.60     Amount  for  2  yr.,  or  3d  principal. 

67.416  Interest  of  $1123.60  for  1  yr.  

1191.016  Amount  for  3  yr.  1191.016 

1000      _  Original  principal.  1000 

191.016  Compound  interest  for  3  yr.  191.016 

482.  RULE. — Find  the  amount  of  the  given  prineipal 
for  the  first  period  of  time,  and  make  it  the  principal  for 
the  second.    Find  the  amount  of  the  second  principal  for 
the  second  period  of  time,  and  make  it  the  principal  for 
the  third;  and  so  continue  for  the  whole  time.    The  last 
amount  is  the  amount  required. 

The  last  amaunt,  less  the  given  principal,  will  be  the 
compound  interest. 

NOTES. — 1.  When  the  time  is  not  a  multiple  of  the  interest  period,  find 
the  amount  of  the  principal  to  the  end  of  the  last  period ;  then  compute  the 
simple  interest  on  this  amount  for  the  remaining  time,  and  add  it  to  the  last 
amount.  The  sum  will  be  the  required  amount. 

2.  The  work  of  computing  compound  interest  may  be  shortened  by  using 
the  tables  on  pages  194  and  195. 

*  For  Annual  Interest,  eee  page  319. 


194 


INTEREST. 


[Art.  483 


483.  Table  showing  the  sum  to  which  $1  will  increase,  at  compound 
interest,  in  any  number  of  years  not  exceeding  45. 


Yrs. 

H. 

2J*. 

,.«, 

8#. 

4fo 

4J*. 

5*. 

61 

9ft 

Yrs. 

1 

1.0200 

1.0250 

1.0300 

1.0350 

1.0400 

1.0450 

1.0500 

1.0600 

1.0700 

1 

2 

1.0404 

1.0506 

1.0609 

1.0712 

1.0816 

1.0920 

1.1025 

1.1236 

1.1449 

2 

3 

1.0612 

1.0769 

1.0927 

1.1087 

1.1249 

1.1412 

1.1576 

1.1910 

1.2250 

g 

4 

1.0824 

1.1038 

1.1255 

1.1475 

1.1699 

1.1925 

1.2155 

1.2625 

1.3108 

4 

5 

1.1041 

1.1314 

1.1593 

1.1877 

1.2167 

1.2462 

1.2763 

1.3382 

1.4026 

5 

6 

1.1262 

1.1597 

1.1941 

1.2293 

1.2653 

1.3023 

1.3401 

1.4185 

1.5007 

6 

7 

1.1487 

1.1887 

1.2299 

1.2723 

1.3159 

1.3609 

1.4071 

1.5036 

1.6058 

7 

8 

1.171? 

1.2184 

1.2608 

1.3168 

1.3686 

1.4221 

1.4775 

1.5988 

1.7182 

8 

9 

1.1950 

1.2489 

1,3048 

1.3629 

1.4233 

1.4861 

1.5513 

1.6895 

1.8385 

9 

10 

1.2190 

1.2801 

1.3439 

1.4106 

1.4802 

1.5530 

1.6289 

1.7908 

1.9672 

10 

11 

1.2434 

.3121 

1.3842 

1.4600 

1.5395 

1.6229 

1.7103 

1.8983 

2.1049 

11 

12 

1.2682 

.3449 

1.4258 

1.5111 

1.6010 

1.6959 

1.7956 

2.0122 

2.2522 

12 

18 

1.2936 

.3785 

1.4685 

1.5640 

1.6651 

1.7722 

1.8856 

2.1329 

2.4098 

13 

14 

1.3195 

.4130 

1.5126 

1.6187 

1.7317 

18519 

1.9799 

2.2609 

2.5785 

14 

15 

1.3459 

1.4483 

1.5580 

1.6753 

1.8009 

1.9353 

2.0789 

2.3966 

2-7590 

15 

16 

1.3728 

1.4845 

1.6047 

1.7340 

1.8730 

2.0224 

2.1829 

2.5404 

2.9522 

16 

17 

1.4002 

1.5216 

1.6528 

1.7947 

1.9479 

2.1134 

2.2920 

2.6958 

3.1588 

17 

18 

1.4282 

1.5597 

1.7024 

1.8575 

2.0258 

2.2085 

2.4066 

2.8543 

3,3799 

18 

19 

1.4568 

1.5987 

1.7535 

1.9225 

2.1068 

2.3079 

2.5270 

3.0256 

3.6165 

19 

20 

1.4859 

1.6386 

1.8061 

1.9898 

2.1911 

2.4117 

2.6533 

3.2071 

3.8697 

20 

21 

1.5157 

1.6796 

1.8603 

20594 

2.2788 

2.5202 

2.7860 

3.3996 

4.1406 

21 

22 

1.5460 

1.7216 

1.9161 

2.1315 

2.3699 

2.6337 

2.9253 

3.6035 

4.4304 

22 

23 

1.5769 

1.7646 

1.9736 

2.2061 

2.4647 

2.7522 

3.0715 

3.8197 

4.7405 

23 

24 

1.6084 

1.8087 

2.0328 

2.2833 

2.5633 

2.8760 

3.2251 

4.0489 

5.0724 

24 

25 

1.6406 

18539 

2.0938 

2.3632 

2.6658 

3.0054 

3.3864 

4.2919 

5.4274 

25 

26 

1.6734 

1.9003 

2.1566 

2.4460 

2.7725 

3.1407 

3.5557 

4.5494 

5.8074 

26 

27 

1.7069 

1.9478 

2.2213 

2.5316 

2.8834 

3.2820 

3.7335 

4.8223 

6.2139 

27 

28 

1.7410 

1.99C5 

2.2879 

2.6202 

2.9987 

3.4297 

3.9201 

5.1117 

6.6488 

28 

29 

1.7758 

2.0464 

2.3566 

2.7119 

3.1187 

3.5840 

4.1161 

5.4184 

7.1143 

29 

30 

1.8114 

2.0976 

2.4273 

2.8068 

3.  -434 

3.7453 

4.3219 

5.7435 

7.6123 

30 

31 

1.8476 

2.1500 

2.5001 

2.9050 

3.3731 

3.9139 

4.5380 

6.0881 

8.1451 

31 

32 

1.8845 

2.2038 

2.5751 

3.0037 

3.5081 

4.0900 

4.7649 

6.4534 

8.7153 

32 

33 

1.9222 

2.2589 

2.6523 

3.1119 

3.6434 

4.2740 

5.0031 

6.8406 

93253 

33 

34 

1.9607 

2.3153 

2.7319 

3.2209 

3.7943 

4.4664 

5.2533 

7.2510 

9.9781 

34 

35 

1.9999 

2.3732 

28139 

3.3336 

3.9461 

4.6673 

5.5160 

7.6861 

10.6766 

35 

36 

2.0399 

24325 

28983 

3.4503 

4.1039 

4.8774 

5.7918 

8.1473 

11  4239 

36 

37 

2.0807 

2.4933 

2.9852 

8.5710 

4.2681 

5.0969 

6.0814 

8.6361 

12.2236 

37 

38 

2.1223 

2.5557 

3.0748 

3.6960 

4.4388 

5.3262 

6.3855 

9.1543 

13.0793 

38 

39 

2.1647 

2.6196 

3.1670 

3.8254 

4.6164 

5.5659 

6.7048 

9.7035 

13.9948 

39 

40 

2.2080 

2.6851 

3.2620 

3.9593 

4.8010 

5.8164 

7.0400 

10.2857 

14.9745 

40 

41 

2.2522 

2.7522 

3.3599 

4.0978 

4.9931 

6.0781 

7.3920 

10.9029 

16.0227 

41 

42 

2.2972 

2.8210 

3.4607 

4.2413 

5.1928 

6.3516 

7.7616 

1.5570 

17.1443 

42 

43 

2.3432 

2.8915 

3.5645 

4.3897 

5.4005 

6.6374 

8.1497 

12.2505 

18.3444 

43 

44 

2.3901 

2.9638 

3.6715 

4.5433 

5.6165 

C.9361 

8.5572 

2.9855 

19.6285 

44 

45 

2.4379 

3.0379 

3.7816 

4.7024 

5.8413 

7.2482 

8.9850 

13.7646 

21.0025 

45 

To  find  the  sum  to  which  a  given  amount  will  increase,  at  compound  interest,  at  any  of 
the  rates  per  cent,  and  number  of  years  expressed  in  the  above  Table  : 

Multiply  the  given  amount  by  the  sum  to  which  one  dollar  will  increase  at  the  rate  and 
for  the  number  of  years  required,  marking  off  as  many  decimals  from  the  product  as  there 
are  decimals  in  the  multiplier  and  multiplicand. 

NOTES.— 1.  The  amount  for  any  number  of  years  not  given  in  the  table  may  be  computed 
by  finding  the  product  for  any  two  numbers  of  years  whose  sum  equals  the  given  time.  Thus, 
the  compound  amount  of  $1  at  6#  for  55  years,  may  be  found  by  multiplying  $13.7646,  the 
amount  for  45  years,  by  1.7908,  the  amount  for  10  years. 

2.  If  the  interest  is  compounded  semi-annually,  to  find  the  amount  from  the  table,  take 
twice  the  number  of  years  at  one-half  the  rate.    Thus,  the  amount  at  8#,  compounded  semi- 
annually,  for  5  years,  is  equivalent  to  the  amount  for  10  periods  of  6  months  each,  at  4#  for 
each  period,  and  is  the  same  as  the  amount  for  10  years  at  4%.    If  the  interest  is  compounded 
quarterly,  take  4  times  the  number  of  years  at  one-fourth  the  rate. 

3.  The  compound  interest  of  $1  is  $1  less  than  the  amounts  in  the  above  table. 


Ait.  484.] 


COMPOUND     INTEREST. 


195 


484.  Table  showing  the  sum  to  which  $1,  paid  at  the  beginning  of  each 
year  will  increase  at  compound  interest,  in  any  number  of  years  not  exceeding  50. 


Yrs. 

3£. 

8J*. 

4%. 

5%. 

6*. 

*?rf 

8%. 

10*. 

Yrs. 

1 

1.0300 

1.0350 

1.0400 

1.0500 

1.0600 

1.0700 

1  0800    1.1000 

1 

2 

2.0909 

2.1062 

2.1216 

2.1525 

2.1836 

2.2149 

22464 

2.3100 

2 

3 

3.1836 

3.2149 

3.2465 

3.3101 

3,3746 

3.4399 

3.5061 

3.6410 

3 

4 

4.3091 

4.3623 

4.4163 

4.5256 

4.6371 

4.7507 

4.8666 

5.1051 

4 

5 

5.4684 

5.5502 

5.6330 

5.8019 

5.9753 

6.1533 

6.3359 

6.7156 

5 

6 

6.6625 

6.7791 

6.8983 

7.1420 

7.3938 

7.6540 

7.9228 

8.4872 

6 

7 

7.8923 

8.0517 

8.2142 

8.5491 

8.8975 

9.2598 

9.6366 

10.4359 

7 

8 

9.1591 

9.3685 

9.5828 

10.0266 

10.4913 

10.9780 

11.4876 

12.5795 

8 

9 

10.4639 

10  7314 

11.0061 

11.5779 

12.1808 

12.8164 

13.4866 

14.9374 

9 

10 

11.8078 

12.1420 

12.4864 

13.2068 

13.9716 

14.7836 

15.6455 

17.5312 

10 

11 

13.1920 

13.6020 

14.0258 

14.9171 

15.8699 

16.8885 

17.9771 

213843 

11 

12 

14.6178 

15  1130 

15.6268 

16.7130 

17.8821 

19.1406 

20.4952 

23.5227 

12 

13 

16.0863 

16.6770 

17.2919 

18.5986 

20.0151 

21.5505 

23.2149 

26.9750 

13 

14 

17.5989 

18.2957 

19.0236 

20.5786 

22.2760 

24.1290 

26.1521 

30.7725 

14 

15 

19.1569 

19.9710 

20.8245 

22.6575 

24.6705 

26.8881 

29.3243 

34.9497 

15 

16 

20.7616 

21.7050 

22.6975 

24.8404 

27.2129 

29.8402 

32.7502 

39,5447 

16 

17 

22.4144 

23.4997 

24.6454 

27.1324 

29.9057 

32.9990 

36.4502 

44.5992 

17 

18 

24.1169 

25.3573 

26.6712 

29.5390 

32.7600 

36.3790 

40.4463 

50.1591 

18 

19 

25.8704 

27.2797 

28.7781 

32.0660 

35.7856 

39.9955 

44.7620 

56.2750 

19 

20 

27.6765 

29.2695 

30.9692 

34.7193 

38.9927 

43.8652 

49.4229 

63.0025 

20 

21 

29.5368 

31.3290 

33.2480 

37.5052 

42.3923 

48.0058 

54.4568 

70.4027 

21 

22   31.45-29 

33.4604 

35.6179 

40.4305 

45.9958 

52.4361 

59.8963 

78.5430 

22 

23 

33.4265 

35.6665 

38.0826 

43.5020 

49.8156 

57.1767 

65.7648 

87.4973 

23 

24 

35.4593 

37.9499 

406459 

46.7271 

53.8645 

62.2490 

72.1059 

97.3471 

24 

25 

37.5530 

40.3131 

43.3117 

50.1135 

58.1564 

67.6765 

78.9544 

108.1818 

25 

26 

39.7096 

42.7591 

46.0842 

53.6981 

62.7058 

73.4838 

86.3508 

120.0999   26 

27 

41.9309 

45.2906 

48.9676 

57.4036 

67.5281 

79.6977 

94.3388 

133.2099 

27 

28 

44.2188 

47.9108 

51.9663 

81.3227 

72.6398 

86.3465 

102.9659 

147.6309 

28 

29 

46.5754 

50.6227 

55.0849 

65.4388 

78.0532 

93.4608 

112.2332 

163.4940 

29 

30 

49.0027 

53-4295 

58.3283 

69.7608 

83.8017 

101.0730 

122.3459 

180.9434 

30 

31 

51.5028 

56.3345 

61.7015 

74.2988 

89.8898 

109.2182 

133.2135 

200.1378 

31 

32 

54.0778 

59.3412 

65.2095 

79.0638 

96.3432 

117.9334 

144.9506 

221.2515 

32 

33 

56.7302 

62.4532 

68.8579 

84.0670 

03.1838 

127.2588 

157.6267 

244.4767 

33 

34 

59.4621 

65.6740 

72.6522 

89.3203 

110.4348 

137.2369 

171.3168 

270.0244 

34 

35 

62.2719 

69.0076 

76.5983 

94.8363 

118.1209 

147.9135 

186.1021 

298.1268 

35 

36 

65.1742 

72.4579 

807022 

100.6281 

26.2681 

159.3374 

202.0703 

329.0395 

36 

37 

68.1594 

76.0289 

84.9703 

106.7095 

134.9042 

171.5610 

219.3159 

363.0434 

37 

38 

71.2342 

79.7249 

89.4091 

113.0950 

144.0585 

184.6403 

237.&412 

400.4478 

38 

39 

74.4013 

83.5503 

94.0255  119.7998 

153.7620 

198.6351 

258.0565 

441.5926 

39 

40 

77.6633 

87.5095 

93.8265 

126.8398 

164.0477 

213.6096 

279.7810 

486.8518 

40 

41 

81.0232 

91.6074 

03.8196 

134.2318 

174.9506 

229.6322 

303.2435 

536.6370 

41 

43 

84.4839 

95.8486 

09.0124 

141.9933 

186.5076 

246.7765 

328.5830 

591.4007 

42 

43 

88.0484 

100.2383  1  114.  4129 

150.1430 

198.7580 

265.1208 

355.9496 

651.6408 

43 

44 

9i.7199 

104.7817 

120.0294 

158.7002 

211.7435  2S4.7493 

385.5056 

717.9048 

44 

45 

95.5015 

109.4840 

125.8706 

167.6852 

225.5081 

305.7518 

417.4261 

790.7953 

45 

46 

99.3965 

114.3510 

131.9454 

177.1194 

240.0986 

328.2244 

451.9002 

870.9749 

46 

47 

103.4084 

119.3883 

138.2632 

187.0254 

255.5645  1  352.2701 

489.1322 

959.1723 

47 

48 

107.5406 

124.6018 

144.8337 

197.4267 

271.9584  377.9990 

529.3427 

1056-1896 

48 

49 

111.7969 

129.9979 

151.6671 

208.3480 

289.3359  405.5289 

572.7702 

1162.9085 

49 

50 

116.1807 

135.5828 

158.77-38 

219.8154 

307.7561  434.9859 

619.6718 

1280.2993 

50 

To  find  the  sum  to  which  a  given  amount,  per  annum,  will  increase  at  compound  inter- 
est, at  any  of  the  rates  per  cent,  and  number  of  years  expressed  in  the  above  Table : 

Multiply  the  given  amount,  per  annum,  by  the  sum  to  which  one  dollar  per  annum  win 
increase  at  the  rate  and  for  the  number  of  years  required,  marking  off  as  many  decimals  from 
the  product  as  there  are  decimals  in  the  multiplier  and  multiplicand. 

NOTE.— If  the  amount  be  payable  semi -annually,  and  compound  interest  is  to  be  allowed 
Bemi-annually,  take  the  amount  for  double  the  number  of  years  at  one-half  the  rate  per  cent. 
Thus,  for  a  semi-annual  payment  of  $1  for  10  years  at  10  per  cent.,  take  the  amount  of  $1  for 
20  years  at  5  per  cent.  =  $34.7193.  For  a  quarterly  payment,  take  the  amount  for  four  times 
the  number  of  years  at  one -fourth  the  rate  per  cent. 


19G  INTEREST.  [Art.  485. 


EXAMPLES. 

485.  1.  What  will  $450  amount  to  at  compound  interest,  in 
4  years,  compounded  annually  at  4%  ?  At  3%? 

2.  Find  the  compound  interest  of  $360,  for  2  years,  interest 
compounded  semi-annually  at  6%.     At  b%. 

3.  AVhat  is  the  compound  interest  of  $800  for  1  yr.  3  mo.  at 
8%,  interest  compounded  quarterly  ? 

4-  At  compound  interest,  what  is  the  amount  of  $1728  for  3  yr. 
4  mo.  16  da.,  interest  compounded  annually  at  Z%  ?  At  §%  ? 

NOTE. — First  find  the  amount  for  3"  years,  and  use  this  amount  as  the 
principal  for  the  remaining  time. 

5.  B  holds  a  mortgage  against  A's  property  dated  Apr.  1,  1881, 
for  $20000,  interest  payable  annually  at  6%.     The  interest  due 
Apr.  1,  1882,  is  not  paid  until  May  26,  1882.     How  much  is  then 
due,  A  having  consented  to  pay  interest   upon  interest  ?     (See 
Note  2,  Art.  481.) 

NOTE. — In  solving  the  following  examples,  use  the  tables  in  Art.  483- 
484. 

6.  A  gentleman   deposits   in   a   savings  bank  $100  when  his 
child  is  one  year  old.     How  much  will  this  amount  to  when  the 
child  is  21  years  old,  interest  being  compounded  semi-annually 
at  4^?    At  5^? 

7.  If,  at  the  age  of  25  years,  a  person  places  $2000  on  interest, 
compounded  annually  at  6^,  what  will  be  the  amount  due  him 
when  he  is  50  years  old  ? 

8.  What  will  $625  amount  to  at  compound  interest,  in  36  years, 
compounded  annually  at  3^  ?    At  4%  ? 

9.  At  the  age  of  20,  and  every  year  thereafter,  a  young  man 
places  $200  at  compound  interest  at  6%.     How  much  will  he  have 
at  the  age  of  30  ?    At  the  age  of  40  ?     (See  Art.  484.) 

10.  How  much  will  a  gentleman  have  at  the  end  of  three  years, 
if  he  places  at  compound  interest  at  5%  $300  at  the  beginning  of 
each  year  ? 

11.  Mr.  B.,  whose  life  is  insured  for  $4000,  pays  an  annual 
premium  of  $114.     How  much  would  this  amount  to  at  6%  com- 
pound interest  in  20  years  ? 

12.  A  lady  deposits  $50  in  a  savings  bank  Jan.  1  and  July  1, 
of  each  year  ;  how  much  will  be  placed  to  her  credit  in  15  years, 
money  being  worth  §%,  compound  interest  ? 


Art.  485.]  COMMERCIAL     PAPER.  197 

IS.  What  sum  must  be  placed  at  compound  interest,  at  6$,  to 
amount  to  $1000  in  5  years  ? 


.-^In  compound  interest,  as  in  simple  interest,  the  amounts  are 
proportional  to  the  principals  ;  hence  the  amount  of  any  principal  is  as  many 
times  greater  than  the  amount  of  $1,  as  that  principal  is  greater  than  $1. 

To  find  the  principal,  divide  the  given  amount  by  the  amount  of  $1  for 
the  given  time  and  rate. 

In  simple  interest,  the  interest  on  a  given  principal  for  a  given  time  is  in 
proportion  to  the  rate  per  cent.,  and  at  a  given  rate,  in  proportion  to  the  time; 
but,  in  compound  interest,  such  is  not  the  case.  If  the  rate  or  time  be  doubled, 
the  interest  is  more  than  doubled. 

14.  How  much,  should  a  gentleman  invest  at  compound  inter- 
est, 6$,  for  his  son  who  is  now  6  years  old,  so  that,  when  he  be- 
comes 21  years  of  age,  he  may  have  $10000  ? 

15.  In  the  above  example,  how  much  should  be  invested  at  the 
beginning  of  each  year  to  produce  the  same  sum  ? 

16.  A  gentleman  at  his  death  left  $7850  for  the  benefit  of  his 
only   son,    12   years  old,   the  money  to  be  paid  to  him  when  he 
should  be  21  years  of  age.     How  much  did  he  receive,  interest  at 
6$,  compounded  send-annually  ?  /' 

17.  How  much  must  a  person  at  the  age  of  25  years,  place 
compound  interest  at  6%,  so  that  the  amount  due  him,  when  h 
is  50  years  old,  will  be  $20000  ? 

18.  In  the  above  example,  how  much  should  he  invest  anntfe 
to  produce  the  same  sum  ? 


COMMERCIAL     PAPER. 

486.  Commercial   Paper  embraces  notes,  drafts,  bil 
exchange,  etc. 

487.  A  Note    (also  called  a  Promissory  Note)  is  a -. 
promise  to  pay  a  certain  sum  of  money  on  demand  or  at  a  s 
time. 

488.  The  Maker  of  a  note  is  the  person  who  signs  it,  and 
thus  becomes  responsible  for  its  payment.     The  Payee  is  the 
person  to  whom,  or  to  whose  order,  it  is  made  payable.     The 
Face  of  a  note  is  the  sum  promised. 

In  Note  1,  Art.  495,  Peter  Cooper  is  the  maker  ;  George  Peabody  is  the 
payee;  the  face  of  the  note  is  $1000. 


198  INTEREST.  [Art.  489. 

489.  A  Negotiable  Note  is  a  note  which  is  made  payable  to 
bearer  or  to  the  order  of  some  person.     (See  Notes,  Art.  495.) 

1.  A  note  is  non-negotiable  when  it  is  payable  only  to  the  party  named  in 
the  note. 

2.  A  negotiable  note  made  in  New  Jersey  must  contain  the  words  "  with- 
out defalcation  or  discount  ; "  in  Missouri,  the  words  ' '  negotiable  and  payable 
without  defalcation  or  discount." 

3.  Negotiable  notes  payable  to  order  may  be  sold  or  transferred  by  the 
payee  writing   his  name  upon  the  back  of  the  note.     He  then  becomes  an 
indorser. 

4.  Negotiable  securities  are  good  in  the  hands  of  one  who  purchases  in 
good   faith   and   before   maturity,  although   the  seller  may  have   found  or 
stolen  them. 

5.  Where  no  place  of  payment  is  specified,  a  promissory  note  is  payable 
at  the  maker's   place  of   business,   or  if  none  is  known,  at  the  residence  of 
the  maker. 

6.  A  note  or  draft  must  be  presented  at  the  place  where  it  is  made  pay- 
able.    If  at  a  bank,  during  banking  hours ;  if  at  a  place  of  business,  during 
business  hours  ;  if  at  a  residence,  during  family  hours  ;  and  if  the  maker,  or 
some  one  for  him,  is  not  ready  with  legal  tender  currency  to  pay  it,  the  holder 
need  not  call  again.     A  check,  even  if  certified,  is  not  a  legal  tender,  and  may 
be  lawfully  refused. 

490.  The  Indorser  of   a  note  or  draft  is  the  person  who 
writes  his  name  on  the  back  of  it,  and  by  so  doing  guarantees 
its  payment. 

If  Mr.  Erastus  Corning  desires  to  sell  or  transfer  Note  3,  Art.  495,  it 
will  be  necessary  for  him  to  indorse  it.  If  he  writes  his  name  only,  it  is  called 
an  indorsement  in  blank,  and  the  note  is  then  payable  without  further  indorse- 
ment to  any  person  lawfully  holding  the  same.  He  may  indorse  it  in  full  by 
making  it  payable  to  a  particular  person,  thus — "Pay  to  the  order  of  Henry 
11.  Pierson,  Erastus  Corning."  Before  it  can  be  again  transferred,  it  will 
require  the  indorsement  of  Henry  R.  Pierson.  For  greater  security,  checks, 
notes,  drafts,  etc.,  are  indorsed  in  full  wrhen  sent  by  mail. 

If  an  indorser  does  not  wish  to  guarantee  the  payment  of  a  note,  draft, 
etc.,  he  writes  "Without  recourse"  over  his  name  at  the  time  of  the  indorse- 
ment. 

Sometimes  notes  and  drafts  are  drawn  to  the  order  of  the  maker  or  the 
drawer  (to  the  order  of  myself  or  ourselves)  to  facilitate  their  transfer  without 
the  indorsement  of  the  holder. 

491.  A  Draft,  or  Bill  of  Exchange  is  an  order  or  request 
addressed  by  one  person  to  another,  directing  the  payment  of  a 
specified  sum  of  money  to  a  third  person  or  to  his  order. 


Art.  492.]  COMMERCIAL     PAPER.  199 

492.  The  Drawer  of  the  draft  is  the  person  who  signs  it. 
The  Drawee  is  the  person  on  whom  it  is  drawn.     The  Payee 
is  the  person  to  whom,  or  to  whose  order,  it  is  made  payable. 

In  Draft  5,  Art.  495,  C.  P.  Huntington  is  the  drawer;  Drexel,  Morgan 

&  Co.  are  the  drawees;  J.  &  W.  Seligman  &  Co.  arc  the  payees. 

1.  The  person  in  whose  favor  the  bill  is  drawn  is  sometimes  called  the 
buyer,  and  becomes  the  "  remitter."     After  the  bill  is  presented  and  accepted, 
the  drawee  is  called  the  acceptor,  and  the  draft,  an  acceptance.     The  draft 
then  has  the  same  legal  significance  as  a  promissory  note. 

2.  A  person  accepts  or  promises  to  pay  a  draft  by  writing  the  word 
"  Accepted  "  and  the  date  over  his  name  across  its  face. 

3.  Drafts  are   sometimes   accepted   in  the  following  form: — "Accepted 
August  20,  1881,  and  payable  at  the  National  Park  Bank,  New  York,  G.  B. 
Horton  &  Co." 

4.  In  the  State  of  New  York,  both  by  law  and  custom,  the  drawee  of  a 
draft  may  demand  24  hours  consideration  from  the  time  the  draft  is  presented 
for  acceptance. 

When  accepted,  it  must  bear  the  date  when  first  seen  by  him. 

5.  To  "honor"  a  draft  is  to  accept  it  or  pay  it  on  being  presented. 

493.  A  Protest  is  a  formal  statement  made  by  a  Notary 
Public,  declaring  that  a  draft  or  note  has  been  presented  for  pay- 
ment or  acceptance,  and  was  refused. 

494.  Days  of  G-race  and  Maturity. — The  day  of  matu- 
rity is  the  day  on  which  a  note  becomes  legally  due.     According 
to  the  laws  of  most  of  the  States,  a  note  is  not  legally  due  until 
three  days  after  the  expiration  of  the  time  specified  in  the  note, 
except  the  note  contain  the  words  "without  grace/'     These  days 
are  called  days  of  grace,  but  they  are  of  no  advantage  to  the 
payer,  since  interest  is  charged  for  them  as  for  any  others. 

1.  California  has  abolished  days  of  grace  altogether.     In  Georgia,  Ala- 
bama, and  Kentucky,  grace  is  allowed  on  promissory  notes  only  in  case  they 
are  made  payable,  or  are  discounted  or  left  for  collection,  at  a  bank  or  private 
banker's. 

2.  The  following  is  an  analysis  of  the  Holiday  law  of  1887  of  the  State  of 
New  York  : 

Paper  due  on  a  holiday  is  payable  the  following  business  day. 

Paper  due  on  Saturday,  except  when  payable  at  sight  or  on  demand,  is  payable  the  fol- 
lowing business  day. 

Paper  due  on  Sunday  must  be  paid  on  the  business  day  following  it.  If  one  of  the  men- 
tioned holidays  falls  on  Sunday,  paper  due  on  that  day  must  be  paid  the  following  business 
day. 

Paper  due  on  Monday,  where  the  preceding  Sunday  is  a  holiday,  is  not  payable  until  the 
following  business  day. 


200  INTEREST.  [Art.  494 

3.  In  nearly  all  of  the  States,  excepting  New  York,  paper  due  on  Sunday, 
or  on  a  legal  holiday,  must  be  paid  the  preceding  business  day.     Thus,  if  a 
holiday  falls  on  Thursday,  all  notes,  etc.,  must  be  paid  on  Wednesday ;  if  a 
holiday  falls  on  Monday,  all  notes  due  Sunday  or  Monday  would  be  payable 
on  Saturday ;  if  a  holiday  falls  on  Saturday,  notes  due  Saturday  or  Sunday 
would  be  payable  on  Friday. 

4.  The  legal  holidays  in  the  State  of  New  York  are  New  Year's  Day  (Jan. 
1),  Washington's  Birthday  (Feb.  22),  Decoration  Day  (May  30),  Independence 
Day  (July  4),  Labor  Day  (the  first  Monday  of  September),  Election  Day  (the 
first  Tuesday  after  the  first  Monday  of  November),  Thanksgiving  Day  (the 
day  appointed  by  the  President  of  the  United  States  and  Governor  of  the 
State,  usually  the  last  Thursday  of  November),  Christmas  Day  (Dec.  25),  and 
every  Saturday  from  12  o'clock  at  noon  until  12  o'clock  at  midnight  (Saturday 
Half-Holiday). 

When  a  legal  holiday  falls  on  Sunday,  Monday  is,  by  the  statute  of  New 
York,  made  a  legal  holiday. 

5.  A  note  made  due  at  a  fixed  date  in  the  future,  carries  3  days'  grace 
(unless  the  words  "without  grace"  are  used  in  the  contract).     Thus,  a  note 
stating  that  "  on  May  1,  1882,  I  promise,  etc.,"  would  carry  3  days'  grace,  and 
would  be  payable  May  4,  1882. 

6.  When  the  time  of  a  no,te  is  expressed  in  months,  calendar  months  are 
used  to  determine  the  day  of  maturity ;  when  in  days,  the  exact  number  of 
days  is  used. 

Thus,  a  note  dated  July  16,  and  payable  two  months  from  date,  would 
nominally  mature  Sept.  16,  and,  including  the  three  days  of  grace,  would 
legally  mature  Sept.  19.  A  note  having  the  same  date,  and  payable  sixty 
days  from  date,  would  nominally  mature  Sept.  14,  and,  including  the  three 
days  of  grace,  would  legally  mature  Sept.  17. 

7.  A  note  due  in  one  or  more  months  from  date,  matures  on  the  corre- 
sponding day  of  the  month  up  to  which  it  is  reckoned,  if  there  are  so  many 
days  in  that  month  ;  but  if  not  so  many,  it  then  matures  on  the  last  day  of 
said  month,  to  which  the  usual  grace  must  be  added.     Thus,  notes  dated  Jan. 
28,  29,  30,  or  31,  and  payable  one  month  from  date,  would  become  due  Mar.  3 
(Feb.  28  with  3  days'  grace  added). 

8.  When  drafts  are  payable  a  certain  time  after  sight,  the  date  of  accept- 
ance and  the  time  of  the  draft  determine  the  day  of  maturity.    Thus,  if  a  draft 
is  dated  May  16,  accepted  May  20,  and  payable  sixty  days  after  sight,  it  would 
mature  or  be  due  63  (including  3  days  of  grace)  days  after  May  20,  or  July  22. 
If  payable  60  days  after  date,  it  would  mature  63  days  after  May  16,  or  July  18. 
It  is  not  necessary  to  present  for  acceptance  drafts  drawn  a  certain  time  after 
date,  but  as  a  courtesy  to  the  drawee,  it  is  usually  done. 

9.  Days  of  grace  are  allowed  on  drafts  according  to  the  custom  of  the 
place  where  they  are  payable.     The  statute  of  New  York  forbids  grace  on  all 
sight  drafts,  no  matter  on  whom  drawn,  and  on  all  time  drafts  which  appear 
on  their  face  to  be  drawn  ' '  upon  any  bank,  or  upon  any  banking  association 
or  individual  banker,  carrying  on  the  banking  business  under  the  act  to 
authorize  the  business  of  banking." 


Art.  495.]      FORMS     OF    NOTES    AND     DRAFTS.  201 

495.  FORMS  OF  NOTES  AND  DRAFTS. 

1.  DEMAND  NOTE. 

$1000.  NEW  YORK,  August  19,  1887. 

On  demand,  I  promise  to  pay  GEORGE  PEABODY,  or  bearer, 

One  Thousand  Dollars.     Value  received. 

PETER  COOPER. 

The  above  note  is  payable  on  demand, — that  is,  whenever  presented;  is 
negotiable  (payable  to  bearer) :  and  bears  interest  from  date  at  the  legal  rate 
of  the  State  in  which  it  is  made.  If  the  words  "  or  bearer"  were  omitted  the 
note  would  not  be  negotiable. 

How  much  would  be  due  on  the  above  note,  Jan.  1,  1888, 
finding  the  time  by  compound  subtraction  ? 

2.  TIME  NOTE — INTEREST-BEARING. 
$876jfo.  CINCINNATI,  OHIO,  July  16,  1888. 

Six  months  after  date,  I  promise  to  pay  GEO.  C.  MILLER, 
or  order,  Eight  Hundred  Seventy-five  and  -ffa  Dollars,  with 
interest  at  eight  per  cent.  Value  received. 

ALEX.  MCDONALD. 

The  above  note  is  payable  6  mo.  3  da.  after  its  date,  or  Jan.  19,  1889 ;  is 
negotiable  (payable  to  order);  and  draws  interest  from  its  date  at  8fo  per 
annum.  If  the  rate  of  interest  was  omitted,  it  would  bear  interest  at  the 
legal  rate  of  the  State  for  such  cases,  6$.  (See  Art.  436.) 

How  much  would  be  due  on  the  above  note  at  its  maturity  ? 
How  much,  March  1,  1889  ?  Supposing  the  rate  of  interest  to  be 
omitted  in  the  note,  how  much  would  be  due  May  4,  1889  ? 

3.  TIME  NOTE — WITHOUT  INTEREST — PAYABLE  AT  A  BANK. 
$6000.  ALBANY,  N.  Y.,  December  4,  1889. 

Sixty  days  after  date,  I  promise  to  pay  to  the  order  of  ERASTUS 
CORNING,  Six  Thousand  Dollars,  at  the  Second  National  Bank. 

Value  received. 

DAVID  MURRAY. 

The  above  note  is  payable  63  days  from  Dec.  4,  1889,  or  Feb.  "5,  1890.  It 
is  payable  at  the  Second  National  Bank.  No  interest  will  be  due  at  maturity 
(Feb.  5).  If  the  note  is  not  paid  at  maturity,  it  will  bear  interest  from  that 
date. 

Supposing  the  above  note  was  payable  90  days  from  date,  what 
would  be  its  due  date  (311,  Ex.  10)?  The  note  as  given  not  being 
paid  at  maturity,  how  much  would  be  due  Feb.  26,  1890,  protest 
fees  $2.10? 


202  INTEREST.  [Art.  495. 

4.  JOINT  AND  SEVERAL  NOTE. 

$416ffo.  WORCESTER,  MASS.,  May  27,  1888. 

Four  months  after  date,  we  jointly  and  severally  promise  to 
pay  JOHK  S.  BALLARD  &  Co.,  or  order,  Four  Hundred  Sixteen 
-ffo  Dollars,  with  interest  from  date,  value  received. 

T.  K.  EARLE. 
CHAS.  W.  SMITH. 

If  the  above  note  were  written  "we  jointly  promise,  etc.,"  it  would  be 
called  a  joint  note.  The  makers  of  a  joint  note  must  be  sued  jointly,  each 
being  responsible  for  one-half  of  the  amount  of  the  note.  The  makers  of  a 
joint  and  several  note  may  be  sued  separately,  either  being  responsible  for 
the  full  amount  of  the  note. 

How  much  would  be  due  on  the  above  note,  Dec.  30,  1888  ? 
How  much  Sept.  30,  1888,  the  date  of  maturity  ? 

5.  SIGHT  DRAFT. 

$8000.  SAN-  FRANCISCO,  CAL.,  May  1,  1882. 

At  sight,  pay  to  the  order  of  J.  &  W.  SELIGMAN  &  Co.,  Eight 
Thousand  Dollars,  value  received. 

C.    P.  HUNTINGTON. 

To  DREXEL,  MORGAN  &  Co.,  New  York. 

6.  TIME  DRAFT. 

$5000.  BURLINGTON,  IOWA,  June  18,  1887. 

At  sixty  days'  sight,  pay  to  the  order  of  ADDISOK  BALLARD, 
Five  Thousand  Dollars,  value  received,  and  charge  to  account  of 

A.  G.  ADAMS. 

To  BARTON  &  JONES,  Chicago,  111. 

Drafts  are  sometimes  drawn  a  certain  number  of  "  days  after  date."  (See 
Art.  494,  Note  8.) 

For  Foreign  Bills  of  Exchange,  see  Art.    555. 

If  the  above  draft  was  accepted  June  19,  1887,  what  was  the 
date  of  maturity? 

7.  A  sixty-day  (63)  day  note  given  on  Monday  will  mature  on 
what  day? 

8.  A  note  payable  90  (93)  days  from  date  and  given  on  Thurs- 
day will  fall  due  on  what  day  of  the  week  ?     If  payable  in  30  (33) 
days  and  given  on  Friday,  on  what  day  will  it  become  due  ? 

9.  A  note, dated  July  22,  and  payable  in  90  (93)  days,  would 
mature  on  what  date  ? 


Art.  496.]  BANK     DISCOUNT.  203 


BANK    DISCOUNT. 

496.  Bank  Discount  is  simple  interest  of  a  note,  paid  in 
advance,  for  the  number  of  days  the  note  has  to  run.     It  may  be 
computed  by  any  of  the  methods  given  for  simple  interest. 

On  notes  without  interest  (the  usual  case  of  notes  discounted  at  banks), 
bank  discount  is  reckoned  on  their  face,  the  amount  due  at  maturity ;  on  notes 
with  interest,  it  is  reckoned  on  the  amount  due  at  maturity,  or  their  face  plus 
the  interest  for  the  full  time  of  the  note. 

497.  The  Proceeds  of  a  note  is  the  amount  received  by  the 
holder  from  the  bank  when  the  note  is  discounted.      It  is  the 
amount  on  which  the  discount  is  reckoned  less  the  discount. 

498.  Call   Loans. — Banks   in   the  large   cities  loan  large 
amounts   of   money   upon   stocks,   bonds,    negotiable   warehouse 
receipts  for  grain,  cotton,  petroleum,  etc.,  as  collateral  security, 
payable  on  demand  or  on  giving  one  day's  notice.     Such  loans  are 
termed  "  call "  or  demand  loans,  and  interest  on  them  is  paid  at 
the  end  of  the  time.     (See  Art.  436,  Note  d.) 

499.  The  time  to  be  reckoned  on  a  loan  or  note  is  exclusive 
of  the  day  of  date,  but  includes  the  day  of  maturity  or  payment. 
Thus,  in  discounting,  a  note  in  the  City  of  New  York,  Apr.  4, 
which  would  mature  Apr.  24,  the  discount  would  be  calculated 
for  20  days. 

1.  In  Philadelphia,  Baltimore  and  some  other  cities  it  is  the  custom  of 
banks  in  finding  time  to  include  both  the  day  of  discount  and  the  day  of 
maturity.     Thus,  the  discount  on  the  above  note  would  be  reckoned  for  21 
days. 

2.  Banks  of  the  City  of  New  York  reckon  discount  both  on  the  basis  of 
360  and  365  days  to  the  year,  the  greater  number  on  the  former  basis. 

3.  When  notes  are  payable  in  other  cities  or  towns,  some  banks  charge 
interest  for  the  time  required  for  the  collecting  bank  to  remit  the  money,  in 
addition  to  the  interest  on  the  note  from  the  day  of  discount  to  the  legal  day 
of  maturity.     Thus,  if  a  note,  maturing  June  10  and  payable  at  a  Chicago 
bank,  is  discounted  at  a  bank  in  New  York,  the  remittance  in  settlement 
would  not  be  received  before  June  12  in  New  York,  and  the  New  York  bank 
in  discounting  the  note  would  be  justified  in  charging  interest  for  two  days 
beyond  the  day  of  maturity. 

4.  Some  banks  charge  a  small  fee  for  collection  and  exchange  in  addition 
to  the  interest  in  discounting  notes  which  are  payable  in  other  cities  or 
towns. 


204 


INTEREST. 


[Art.  5OO. 


EXAMPLES. 

5OO.  Find  the  date  of  maturity  and  proceeds  of  the  following 
notes : 

(*•) 

610000.  NEW  YOKE,  July  16,  1889. 

Four  months  after  date,  I  promise  to  pay  to  the  order  of  FISK 
&  HATCH,  Ten  Thousand  Dollars,  at  the  first  National  Bank, 
value  received. 

S.  D.  BABCOCK. 

Discounted  July  16,  1889,  at  6$. 

ANALYSIS. — The  note  is  due  4  months  (494,  6)  and  3  days  (days  of  grace, 
494)  after  July  16,  or  Nov.  19.  From  the  day  of  discount  (July  16)  to  the 
day  of  maturity  (Nov.  19)  there  are  126  days. 

The  interest  of  $10000  for  126  days  at  6$,  if  reckoned  on  the  basis  of 
360  days  to  the  year,  is  $210,  and  the  proceeds  are  $10000  less  $210, 
or  $9790. 

The  interest  on  the  basis  of  365  days  to  the  year  would  be  $2.88  less,  or 
$207.12,  and  the  proceeds  would  be  $9792.88. 

If  the  note  was  discounted  Sept.  1,  the  interest  or  discount  would  be 
reckoned  for  79  days  (Sept.  1  to  Nov.  19). 


$8000.  BROOKLYN,  N.  Y.,  July  16,  1891. 

Ninety  days  from  date,  I  promise  to  pay  S.  B.  CHITTEKDEK, 
or  order,  Eight  Thousand  Dollars,  value  received. 

A.  A.  Low. 

Discounted  Aug.  31,  1891,  at  §%. 

ANALYSIS.— The  note  is  due  93  days  (494)  after  July  16,  or  Oct.  17. 
Compute  the  discount  for  47  days  (Aug.  31  to  Oct.  17)  on  $8000. 

If  the  note  had  been  discounted  July  16,  the  date  of  the  note,  the  interest 
would  have  been  computed  for  93  days,  the  full  time  of  the  note. 

NOTE. — The  results  of  the  following  examples  will  be  given  on  the  basis  of 
both  360  and  365  days  to  the  year. 


No. 

Date  of  Note. 

Time. 

Face. 

Date  of  Discount. 

Rate  of 
Discount. 

3 

Jan.  24  

90  days 

$1200 

Jan.  24  

Q% 

4 

May  18.     . 

3  mo. 

$5280 

May  18  

6% 

5 

Aug  31  ...     . 

60  days 

$2560 

Aug.  31  

8% 

6 

June  4  

4  mo. 

$3756 

June  4  

1% 

7 

Oct  16 

30  days 

$6425 

Oct.  16  

§% 

8 

Mar    13 

6  mo 

$8375 

Mar.  13  

51% 

Art.  500.] 


BANK    DISCOUNT. 


205 


No. 

Date  of  Note. 

Time. 

Pace. 

Date  of  Discount. 

Rate  of 
Discount. 

9 

May  29. 

8  mo. 

$4500 

July  7.  

10% 

10 

July  27 

60  days 

$8240 

Sept.  2  

6% 

11 

Mar  28 

90  days 

$4324 

Apr.  14  

51% 

12 

May  27  

6  mo. 

$4885 

Aug.  15  

8% 

13 

Jan.  3  

120  days 

$9000 

Feb.  28  

6% 

14 

Sept   12 

4  mo 

$5000 

Oct   14 

7% 

15 

Nov   1 

90  days 

$6000 

Nov  28 

5-4-<& 

U5  7° 

16.  What  were  the  proceeds  of  Note  3,,  Art.  495,  if  discounted 
Dec.  16,  1889,  at  the  legal  rate  ? 

17.  Find  the  date  of  maturity  and  proceeds  of  a  note  of  $5000 
payable  60  days  from  date,  dated  and  discounted  at  a  Philadelphia 
bank,  Aug.  3.     (See  Art.  499,  1.) 

18.  Find  the  date  of  maturity  and  proceeds  of  a  note  of  $3750, 
payable  60  days  from  date,  dated  and  discounted  at  a  Maryland 
bank,  Jan.  31,  1882. 

NOTE. — In  the  following  examples,  the  charge  for  collection  and  exchange 
is  a  certain  per  cent,  of  the  face  of  the  notes,  without  reference  to  the  time 
they  have  to  run  (499,  4). 


No. 

Date  of  Note. 

Time. 

Face. 

Date  of  Discount. 

Rate  of 
Discount. 

Rate  of 
Collection. 

19 

May  5 

90  days 

$7000 

May  5 

6% 

1% 

20 

March  1.  ... 

4  mo. 

$9000 

Apr.  30  

*% 

-&% 

21 

June  18  

6  mo. 

$5000 

July  31  

Sf0 

k% 

22 

July  28.... 

60  days 

$4500 

Aug.  1  

e# 

-h% 

23 

Sept.  3  

90  days 

$9000 

Sept.  5  

5|£ 

<h% 

24 

Aug.  5  

4  mo. 

$3000 

Oct.  18  

6^ 

\% 

Required  the  proceeds  and  date  of  maturity  of  the  following 
notes  discounted  (360  days  to  the  year)  through  a  broker,  his 
commission  being  \%  of  the  face  of  the  notes. 


No. 

Date  of  Note. 

Time. 

Face. 

Date  of  Discount. 

Rate  of 
Discount. 

25 

Feb.  21  

4  mo. 

$10000 

Feb.  21  

44% 

26 

June  8  

4  mo. 

$6000 

June  12  

Wo 

27 

Jan.  10  

4  mo. 

$6000 

Jan.  10  

41% 

28 

Mar.  3  

6  mo. 

$8775 

Apr.  30  

4£% 

206  INTEREST.  [Art.  50O. 

29.  May  4,  a  New  York  bank  discounts,  at  6$,  a  note  of  $8000, 
payable  in  St.  Louis,  Sept.  1.     What  are  the  proceeds,  if  interest 
is  charged  for  the  two  days  required  for  the  remittance  of  a  draft 
in  settlement  ? 

30.  A  note  dated  Mar  27,  for  $9000,  payable  in  4  months  at  a 
bank  in  Omaha,  is  discounted  the  same  date  at  a  bank  in  Provi- 
dence,  R.  I.     What  were  the  proceeds,   if   interest   for  3  days 
beyond  the  day  of  maturity  is  charged  ?  < 

81.  A  broker  discounts  a  note  payable  in  4  months  at  4J%, 
and  charges  \%  brokerage.     This  is  equivalent  to  what  rate  of 
interest  per  annum,  making  no  allowance  for  the  days  of  grace  ? 

82.  A  merchant  can  discount  a  note  at  his  bank  at  6%,  365 
days  to  the  year,  or  through  a  broker  at  4f%  360  days  to  the  year, 
broker's  commission  \%.     How  much  better  is  the  latter  method 
on  a  note  of  $10000,  payable  in  4  months,  dated  and  discounted 
May  21  ? 

Find  the  date  of  maturity  and  proceeds  of  the  following 
interest-bearing  notes 

(33.) 
$3000.  ALBANY,  N.  Y.,  September  16,  1881. 

Four  months  after  date,  1  promise  to  pay  W.  J.  KLINE  or 
order,  Three  Thousand  Dollars,  with  interest  at  5%,  value 
received. 

J.  M.  THOMAS. 
Discounted  Nov.  3,  1881,  at  6%. 

NOTE. — Compute  the  discount  at  Q%  for  77  days  (Nov.  3  to  Jan.  19)  on 
the  amount  due  at  maturity  ($3000  plus  the  interest  of  $3000  for  4  months 
and  3  days  at  5  ft). 

34..  A  note  dated  May  27,  1879,  payable  in  3  months,  for 
$3750,  with  interest  at  1% ;  discounted  May  27,  1879,  at  8$. 

35.  A  note  dated  Jan.   16,  1879,  payable  in  4  months,   for 
$1632,  with  interest  at  §% ,  discounted  Mar.  5,  1879,  at  1%. 

36.  A  note  dated  Oct.  12,  1878,  payable  in  6  months,  for  $875, 
with  interest  at  1% ;  discounted  Jan.  10,  1879,  at  10%. 

37.  For  what  amount  must  a  note  be  given  for  60  days  to 
afford  $1000  proceeds,  if  discounted  at  6$  ? 

ANALYSIS. — The  proceeds  of  any  note  is  as  many  times  greater  than  the 
proceeds  of  $1,  as  the  face  of  the  note  is  greater  than  $1.  If  a  note  of  $1  is 
discounted  for  63  days  at  6$,  it  will  afford  $.9895  proceeds;  to  afford  $1000 
proceeds,  the  face  of  the  note  must  be  as  many  times  $1,  as  $.9895  is  con- 
tained times  in  $1000,  or  $1010.61. 


Art.  500.]  PARTIAL     PAYMENTS.  207 

The  following  approximate  method  is  generally  used  by  business  men: 
To  the  given  proceeds,  add  the  interest  for  the  given  time. 

The  interest  of  $1000  for  63  days  is  $10.50.  $1000  +  $10.50  =  $1010.50. 
Since  the  interest  is  reckoned  on  the  proceeds  instead  of  the  face  of  the  note, 
the  error,  11  cents,  is  equivalent  to  the  interest  of  the  interest  ($10.50)  for  the 
given  time. 

Where  greater  accuracy  is  required,  the  necessary  correction  may  be 
made.  The  interest  of  $10.50  for  63  days  is  11  cents.  $1010.50 +$.11 
=  $1010.61. 

88.  A  owes  B  $1500  ;  how  large  a  90-day  note  must  A  give  B 
that  when  discounted  at  a  bank  at  Q%,  the  proceeds  will  be  suffi- 
cient to  pay  the  debt  ? 

89.  I  hold  a  note  of  $3000  against  Mr.  0.,  which  he  pays  by 
giving  a  new  note  at  90  (93)  days  for  $1500,  and  the  balance,  includ- 
ing the  discount  on  the  new  note,  in  cash.     Required  the  amount 
of  cash  paid. 

40.  A  merchant  having  $8000  to  pay,  gets  a  note  for  $5000, 
that  will  mature  in  40  days,  discounted  at  a  bank  at  6$.  How 
large  a  note  must  he  draw,  payable  in  90  (93)  days,  for  discount  at 
the  same  rate,  that  the  proceeds  of  the  two  notes  may  enable  him 
to  meet  his  payment  ? 

PARTIAL    PAYMENTS. 

501.  Partial  Payments  are  payments  in  part  of  a  note, 
mortgage,  or  other  debt,  made  at  different  times. 

502.  Indorsements  are  the  acknowledgments  of  the  pay- 
ments, written  on  the  back  of  the  note,  mortgage,  etc.,  and  stating 
the  amount  and  date  of  the  payment. 

Special  receipts  are  sometimes  given  for  such  payments. 

UNITED    STATES    RULE. 

503.  Ex.     How  much  would  be  due  Sept.  1,  1882,  on  a  note 
of  $600,  dated  March  1,  1882,  with  interest  at  6%  ?     Suppose  a 
payment  of  $100  be  made  Sept.  1,  1882,  to  pay  the  interest  and 
part  of  the  principal,  how  much  would  then  be  due  ?    Ans.  $518. 

Ex.  How  much  would  be  required  to  settle  the  above  note 
Jan.  1,  1883,  the  balance  of  $518  remaining  on  interest  at  the 
same  rate  from  Sept.  1,  1882  ?  Ans.  $528.36. 


208  INTEREST.  [Art.  503. 

Ex.  Find  the  amount  due  on  the  following  note,  Jan.  19, 
1885: 

$1000.  BOSTOK,  MASS.,  Aug.  1,  188  L 

One  year  after  date,  I  promise  to  pay  JORDAN,  MARSH  &  Co., 
or  order,  One  Thousand  Dollars,  for  value  received,  with  interest 
from  date,  at  6  per  cent. 

ALEXANDER  H.  EICE. 

On  this  note  are  the  following  indorsements : 
Received  Apr.  21,  1882,  $200.  Received  Aug.  1,  1883,  $100. 

Received  Dec.  1,  1882,  $25.  Received  July  7,  1884,  $400. 

NOTE. — The  method  given  in  the  following  operation,  is  that  adopted  by 
the  Supreme  Court  of  the  United  States,  and  has  been  made  the  legal  method 
of  nearly  all  the  States.  By  the  United  States  Rule,  as  this  is  generally 
called,  settlements  are  made  whenever  the  payments  are  equal  to  or  exceed 
the  interest  due ;  if  the  payment  exceeds  the  interest,  it  is  applied  first  to 
discharge  the  interest,  and  the  surplus  is  applied  towards  paying  the  princi- 
pal ;  if  the  payment  is  less  than  the  interest,  it  is  not  applied  until  the 
payments,  taken  together,  are  sufficient  to  pay  all  interest  due ;  since  no 
unpaid  interest  is  added  to  the  principal  to  draw  interest,  a  new  principal  can 
never  be  greater  than  the  preceding  principal. 

OPERATION. 

Face  of  note,  or  principal,  from  Aug.  1, 1881        ....        $1000 
Interest  from  Aug.  1, 1881,  to  Apr.  91,  1882  (8  mo.  20  da.),  added  43.33 

Amount,  Apr.  21, 1882, 1043.33 

First  payment,  Apr.  21,  1882, 200.00 

New  principal  from  Apr.  21,  1882 843.33 

Interest  of  $843.33  from  Apr.  21, 1882,  to  Dec.  1,  1882, 

(7  mo.  10  da.) $30.92 

(Interest  exceeds  the  payment,  and  a  new  principal  is 

not  formed.) 
Interest  of  $843.33  from  Dec.  1,  1882,  to  Aug.  1, 1883, 

(8  mo.) 33.73  64.65* 

[Payments  $125  ($25  +  $100),  now  greater  than  the  interest  due 

($64.65)]. 

Amount,  Aug.  1, 1883, 907.98 

Second  and  third  payments,  $25  +  $100 125 

New  principal  from  Aug.  1, 1883 782.98 

*  In  many  cases  it  can  be  determined  mentally  in  advance  whether  the  payment  is 
greater  or  less  than  the  interest.  In  this  case  the  interest  could  be  taken  at  once  from 
Apr.  21, 1882,  to  Aug.  1, 1883  (1  yr.  8  mo.  10  da.),  since  it  is  evident  that  the  payment  ($25)  is 
less  than  the  interest  of  $848.33  for  7  mo.  10  da.  (The  interest  of  $800  for  7  mo.  is  3J  x  $8,  or 
$28,  and  it  would  be  more  on  $843.33  for  7  mo.  10  da.)  If  it  is  doubtful  whether  the  payment 
ia  greater  or  leea  than  the  interest,  perform  all  the  work. 


Art.  503.]  PARTIAL      PAYMENTS.  V*\209 

New  principal  from  Aug.  1,  1883 $782.98 

Interest  of  $782.98  from  Aug.  1, 1883,  to  July  7, 1884  (11  mo.  6  da.)  43.85 

Amount,  July  7,  1884, 826.83 

Fourth  payment,  July  7,  1884, 400 

New  principal  from  July  7,  1884     ..'.....  426.83 

Interest  of  $426.83  from  July  7,  1884,  to  Jan.  19,  1885  (6  mo. 

12  da.) 13.66 

Amount  due  Jan.  19,  1885,  the  final  day  of  settlement,          .        Arts.  $440.49 

504.  UNITED   STATES  EULE. — Find  the  amount    of  the 
given  principal  to  the  time  when  the  payment  or  the  sum 
of  the  payments  exceeds  the  interest  due;  subtract  from 
this  amount  the  payment  or  the  sum  of  the  payments. 
Treat  the  remainder  as  a  new  principal,  and  proceed  as 
before,  to  the  time  of  settlement. 

EXAMPLES. 

505.  NOTES. — 1.  In  the  following  examples,  find  the  time  by  compound 
subtraction. 

2.  In  the  first  five  examples,  all  the  payments  exceed  the  interest. 

$1680.  TRENTON,  N.  J.,  Oct.  9,  1880. 

1.  On  demand,  I  promise  to  pay  COOPER,  HEWITT  &  Co.,  or 
order,  Sixteen  Hundred  Eighty  Dollars.     Value  received. 

JOHN-  A.  ROEBLIKG. 

On  this  note  were  indorsed  the  following  payments  : 
Dec.  21,  1881,  received  $289.12..     June  9,  1883,  received  $991.50, 
How  much  was  due  Jan.  30,  1884  ? 

2.  On  a  note  dated  May  11,  1877,  for  $2000,  are  the  following 
indorsements:— Aug.   6,  1879,  $361;   Feb.  11,   1880,   $901.60; 
Nov.  2,  1882,  $1000.     What  remained  due  Feb.  2,  1883,  at  6%? 
At  5^? 

3.  On  a  note  dated  July  11,  1878,  for  $2400|pe  the  fpllowjng- 
indorsements  :  — Sept.  17,  1879,  $200  ;  Jan.  29M880,  $400  ;  Nov. 
29,1881,  $1150.     What  is  the  amount  due  .din.  11,  1882,  the 
interest  being  at  $%  ?    At  7%  ? 

4.  On  a  mortgage  for  $1700,  dated  May  28, 1880,  there  was 
paid  Nov.  12,  1880,  $80 ;    Sept.   20,  1881,  $314 ;   Jan.  2,  1882, 
$50  ;  Apr.  17,  1882,  $160.     What  was  due  Dec.  12,  1882,  at  6$? 
At 


210  INTEREST.  [Art.  505 - 

6.  On  a  note  dated  May  30,  1879,  for  $1666,  are  the  following 
indorsements:— Apr.  9,  1880,  $314  ;  Nov.  4,  1880,  $180  ;  Aug. 
•  25,  1881,  $575.  What  was  due  June  30,  1882,  at  §%  ?  At  Sfc  ? 

6.  What  was  the  amount  due  Oct.  17,  1881,  upon  a  note  for 
$1000,  dated  New  York,  Mar.  2, 1880,  and  on  which  the  following 
payments  were  indorsed: — June  2,   1880,   $80;    Dec.  15,   1880, 
$20 ;  May  2,  1881,  $32  ;  June  2,  1881,  $60  ? 

7.  A  note  for  $3600,  dated  May  12,  1880,  bore  the  following 
indorsements:— Jan.   2,  1881,  $255  ;  Mar.  15,  1881,  $225  ;  June 
3, 1881,  $120  ;  Aug.  6,  1881,  $300  ;  Feb.  3,  1882,  $30.     What  was 
due  June  2,  1882,  at  6^  ?    At  10%  ? 

8.  A  note  for  $4000,  dated  Mar.  9,  1874,  was  indorsed  as  fol- 
lows:—Jan.   18,  1876,  $300  ;  June  4,  1876,  $400 ;  Dec.  9,  1876, 
$1800  ;  Sept.  1,  1879,  $2000.     How  much  had  to  be  paid  Jan.  1, 
1880,  to  take  up  the  note,  at  $%  ?    At  1%  ? 

9.  A  mortgage  of  $6000  is  dated  May  9,  1877,  on  which  there 
were  the  following  payments: — July  15,  1878,  $500 ;   Nov.  27, 
1878,  $1000 ;  June  1,  1879,  $100 ;  May  9,  1880,  $275  ;  Sept.  27, 
1880,  $2000.     What  was  due  Nov.  9,  1880,  the  interest  being  at 
§%  ?    At  12%  ? 

MERCANTILE     RULES. 

506.  The  following  methods  are  frequently  used  by  merchants 
in  finding  the  balance  due  on  a  note  where  partial  payments  have 
been  made.     They  are  similar  to  the  methods  in  general  use  for 
finding  the  balance  due  on  an  open  account  (592). 

507.  When  the  note  runs  for  one  year  only,  or  less. 

508.  RULE. — Compute  the   interest  on   the  principal 
from  the  time  it  commenced  to  draw  interest,  and  on  each 
payment  from  the  time  it  was  made  until  the  time  of 
settlement,  and  deduct  the  amount  of  all  the  payments, 
including  interest,  from  the  amount  of  the  principal  and 
interest. 

NOTES. — 1.  This  rule  is  used  by  some  merchants  when  the  note  runs 
more  than  one  year,  although  it  is  greatly  to  the  disadvantage  of  the  creditor, 
or  holder  of  the  note. 

2.  In  solving  examples  by  this  rule,  the  different  methods  for  finding 
time  and  interest,  given  in  Art.  4:37*  are  used.  The  results  of  the  following 
examples  will  be  given  for  the  first  method  (Compound  Subtraction  and 
360  days  to  the  year). 


Art.  509.]  PARTIAL     PAYMENTS.  211 


EXAMPLES. 

5O9.  1.  According  to  the  mercantile  rule,  find  the  balance 
due  May  12,  1882,  on  a  note  for  $2400,  dated  July  12,  1881,  on 
which  the  following  payments  have  been  made  :  Dec.  16,  1881, 
$40  ;  Jan.  2,  1882,  $100  ;  Mar.  15,  1882,  $150. 

OPERATION. 

Face  of  note,  or  principal,  July  12,  1881,         .        .        .        .        .      $2400.00 

Interest  on  the  same  to  May  12,  1882  (10  mo.}          .        .        .        .  120.00 

Amount,  May  12,  1882,     .........        2520.00 

First  payment,  Dec.  16,  1881,  ......      $40,00 

Interest  on  the  same  to  May  12,  1882  (4  mo.  26  da.)        .  .97 

Second  payment,  Jan.  2,  1882,          .....       100.00 

Interest  on  the  same  to  May  12,  1882  (4  mo.  10  da.)        .          2.17 
Third  payment,  Mar.  15,  1882,          .....       150.00 

Interest  on  the  same  to  May  12,  1882  (1  mo.  27  da.)        .          1.42  294.56 

Balance  due  May  12,  1882,       ........      $2225.44 

2.  On  a  note  dated  Jan.  13,  1882,  for  $1234,  are  the  following 
indorsements:—  May  17,  1882,  $234  ;  June  16,  1882,  $345  ;  July 
27,  1882,  $123  ;  Sept.  19,  1882,  $135.     What  remained  due  Nov. 
13,  1882,  at  S%  ?    At  1%  ? 

3.  A  note  for  $1567,  dated  Jan.  14,  1881,  bore  the  following 
indorsements:—  Mar.  11,  1881,  $50  ;  May  13,  1881,  $245  ;  June  19, 
1881,  $374;  Aug.  30,  1881,  $412  ;  Sept.  28,  1881,  $316.40.    What 
was  due  Jan.  1,  1882,  at  6^  ?    At  5fc  ? 

4.  On  a  note  dated  Aug.  17,  1881,  for  $3300,  were  the  follow- 
ing indorsements:—  Dec.   18,   1881,  $320;  Feb.   5,   1882,   $425; 
Apr.  13,  1882,  $550  ;  June  29,  1882,  $630  ;  July  16,  1882,  $375  ; 
Aug.   1,  1882,  $500.     What  amount  was  due  Aug.  17,  1882,  at 


510.  When  the  note  runs  for  more  than  one  year. 

511.  Since  it  is  the  custom  of  merchants   and  bankers  to 
balance  their  accounts  annually,  the  following  method  is  used  by 
them  in  computing  the  balance  due  on  a  note  when  it  runs  more 
than  one  year. 

It  is  equivalent  to  finding  the  balance  due  yearly  by  the  previous  rule,  and 
treating  the  balance  as  a  new  principal.  The  periodical  settlements  are  made 
annually,  semi-annually,  or  quarterly,  depending  upon  the  custom  of  the  mer- 
chant or  banker  in  balancing  his  accounts.  Some  merchants  make  the  end  of 
the  business  year,  Jan.  1  or  July  1,  the  periodical  rest,  or  date  of  settlement 
for  notes  and  accounts. 


212 


INTEREST. 


[Art.  512. 


RULE. — Find  the  amount  of  the  principal  for  one 
year ;  also  of  each  payment  made  during  the  year  from  the 
time  the  payment  was  made  to  the  end  of  the  year  (1  yr. 
from  the  date  of  the  note).  From  the  amount  of  the  prin- 
cipal, subtract  the  sum  of  the  payments,  including  interest. 
With  the  remainder  as  a  new  principal,  proceed  thus  for 
each  entire  year  that  follows,  and  for  the  interval  between 
the  end  of  the  last  year  and  the  final  date  of  settlement. 

NOTE. — When  payments  are  made  yearly  greater  than  the  interest  due, 
this  rule  is  the  same  as  the  New  Hampshire  rule  (675)  for  notes  "with  inter- 
est annually." 


EXAMPLES. 

513.  1.  By  the  above  rule,  find  the  balance  due  Jan.  19, 1885, 
at  6%,  on  a  note  for  $2400  dated  Aug  1,  1881,  on  which  the  fol- 
lowing payments  have  been  made: — Apr.  21,  1882,  $200  ;  Dec.  1, 
1882,  $25  ;  Aug.  1,  1883,  $100  ;  July  7,  1884,  $400.  (Time  by 
Compound  Subtraction. ) 


$200.00 
3.33 


OPERATION. 

Face  of  note,  or  principal,  Aug.  1, 1881, 

Interest  on  the  same  for  1  year,     .... 

Amount,  Aug.  1,  1882, 

First  payment,  Apr.  21,  1882,        .... 
Interest  on  the  same  to  Aug.  1,  1882  (3  mo.  10  da.) 

Balance  and  new  principal,  Aug.  1,  1882, 

Interest  on  the  same  for  1  year, 

Amount,  Aug.  1,  1883, 

Second  payment,  Dec.  1,  1882, $25.00 

Interest  on  the  same  to  Aug.  1,  1883  (8  mo.) .        .        .          1.00 
Third  payment,  Aug.  1, 1883,        .        .        .        .        .       100.00 

Balance  and  new  principal,  Aug.  1,  1883, 

Interest  on  the  same  for  1  year, 

Amount,  Aug.  1,  1884, 

Fourth  payment,  July  7,  1884, $400.00 

Interest  on  the  same  to  Aug.  1,  1884,  (24  da)         .        .  1.60 

Balance  and  new  principal,  Aug.  1,  1884, 

Interest  on  the  same  to  date  of  settlement,  Jan.  19, 1885  (5  mo.  18  da.) 
Balance  due  Jan.  19,  1885, 


$2400.00 

144.00 

2544.00 


203.33 

2340.07 

140.44 

2481.11 


126.00 
2355.11 

141.31 
2496.42 


401.60 

2094.82 

58.65 

$2153.47 


2-9.  Solve  Examples  2-9,  Art.  5O5,  according  to  the  mer- 
cantile  rule. 


RATIO    AND    PROPORTION. 


514.  Ratio  is  the  relation  of  two  numbers  as  expressed  by 
the  quotient  of  the  first  divided  by  the  second.     Thus  the  ratio  of 
6  to  3  is  6-^-3,  or  2. 

1.  There  is  no  ratio  between  quantities  of  different  kinds ;  as  6  bu.  and  3 
ft.     But  a  ratio  exists  between  quantities  of  the  same  kind  though  of  different 
denominations;  as  6  ft.  and  8  in.     To  express  the  ratio  in  such  cases,  the 
quantities  must  first  be  reduced  to  the  same  denomination.    Thus,  the  ratio  of 
6  ft.  to  8  in.  is  72  in.-i-S  in.,  or  9. 

2.  The  ratio  between  two  numbers  is  denoted  by  placing  a  colon  (the  sign 
of  division  without  the  horizontal  line)  between  them.     Thus,  the  ratio  of  6 
to  3  is  expressed  6 : 3. 

515.  A  Simple  Ratio  is  a  ratio  between  two  numbers ;  as 
4:5. 

516.  A  Compound  Ratio  is  a  ratio  formed  by  the  combina- 
tion of  two  or  more  simple  ratios. 

Thus,  o !  o  is  a  compound  ratio,  and  is  equivalent  to  4  x  3 : 5  x  2,  or  12: 10. 

517.  The  numbers  whose  ratio  is  expressed  are  the  terms  of 
the  ratio.     The  two  terms  of  a  ratio  form  a  couplet,  the  first  of 
which  is  the  antecedent,  and  the  second,  the  consequent. 

518.  Proportion  is  an  equality  of  ratios. 

The  ratio  of  6  yd.  to  3  yd.  is  2,  and  the  ratio  of  $24  to  $12  is  2 ;  hence 
from  the  two  equal  ratios  the  following  proportion  can  be  formed — 6  yd. :  3  yd. 
—$24:  $12.  This  expression  is  read,  "The  ratio  of  6  yd.  to  3  yd.  equals  the 
ratio  of  $24  to  $12."  In  place  of  the  sign  of  equality  (=),  four  dots  (::)  are 
generally  used;  thus,  6  yd.  :3  yd. : :  $24 :  $12.  The  expression  is  also  read, 
"  6  yd.  is  to  3  yd.  as  $24  is  to  $12." 

519.  The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes ;  and  the  second  and  third,  the  means. 


214  RATIO     AND     PROPORTION.  [Art.  52O. 

520.  PRINCIPLES. — 1.   The  product  of  the  means  is  equal  to 
the  product  of  the  extremes. 

2.  A  missing  mean  may  he  found  hy  dividing  the  product  of 
the  extremes  hy  the  given  mean. 

3.  A  missing  extreme  may  he  found  hy  dividing  the  product  of 
the  means  hy  the  given  extreme. 

SIMPLE     PROPORTION. 

521.  Simple  Proportion  is  an  equality  of  two  simple  ratios; 

as  9  /£.:16/£.::$27:$48. 

Ex.     If  24  hats  cost  $27,  what  will  32  hats  cost  ? 

ANALYSIS. — For  convenience,  make  the  fourth  term  the  missing  term,  or 
the  required  answer.  Since  the  third  and  fourth  terms  must  be  of  the  same 
denomination  and  the  denomination  of  the  answer  will  be  dollars,  take  $27  as 
the  third  term.  From  the  nature  of  the  example,  the  answer  will  be  more 
than  $27,  the  third  term,  therefore  make  32  hats  the  second  term,  and  24  hats 
the  first  term.  The  proportion  will  then  be  stated  as  follows:  24  hats: 32 
hats::  $27: x  (Let  x  represent  the  unknown  term).  Multiplying  32  by  27,  and 
dividing  the  product  by  24,  the  fourth  or  missing  term  will  be  $36. 

522.  RULE. — For  convenience,  take  for  the  third  term 
the  number  that  may  form  a  ratio  with,  or  is  of  the  same 
denomination  as,  the  answer.    If  from  the  nature  of  the 
example,  the  answer  is  to  be  greater  than  the  third  term, 
make  the  greater  of  the  two  remaining  terms  (which  must 
be  of  the  same  denomination)  the  second  term  ;  when  not, 
make  the  smaller  the  second  term.    Then  multiply  the 
means  (the  second  and  third)  together,  and  divide  their 
product  by  the  given  extreme  (the  first  term). 

NOTES. — 1.  After  the  example  is  stated,  any  factor  of  the  given  extreme 
may  be  cancelled  with  an  equal  factor  of  either  of  the  means. 

2.  The  above  rule  is  sometimes  called  the  "  Rule  of  Three." 

EXAMPLES. 

523.  Find  the  missing  term  (represented  by  x)  in  each  of  the 
following  proportions  (See  Principles,  Art.  520). 

1.  16:  x::  24: 18.  5.  $48:  $75  ::  $32: z. 

&  x:  27::  18: 54.  6.  $375:  $144  ::  625  lh.:  x. 

3.  32:27::  x:  135.  7.  $1728:  $288  ::  $666:#. 

4.  24:bu.:32t>u.::$27:x.  8.  144  yd.:  175  yd. ::  $18:  x. 


Art.  523.]  SIMPLE    PROPORTION.  215 

9.  If  19  yd.  of  silk  cost  $28.50,  what  will  37  yd.  cost  ? 

10.  If  64  yd.  of  carpet  36  in.  wide  will  cover  a  floor,  how  many 
yards  27  in.  wide  will  be  required  to  cover  the  same  floor  ? 

11.  A  cane  3  ft.  3  in.  high  casts  a  shadow  5^  ft.  long ;  how  long 
a  shadow  is  cast  by  the  steeple  of  a  church  which  is  234  feet  high  ? 

12.  If  the  freight  of  a  long  ton  (336,  3)  is  70  shillings,  what 
is  the  freight  of  16375  pounds  ? 

13.  The  net  assets  of  a  bankrupt  are  $27675,  and  the  liabilities 
$138375.     How  much  must  be  paid  to  Mr.  A,  whom  he  owes 
$4800  ? 

14.  A  building  is  insured  in  several  companies  for  $28000. 
During  a  fire  the  building  is  damaged  to  the  amount  of  $13500. 
What  is  the  loss  of  company  A,  whose  risk  is  $5000  ? 

15.  A  invests  in  business  $8450,  and  B  $7200,  and  the  gain  or 
loss  is  divided  according  to  the  investments.     "What  is  each  part- 
ner's share  of  gain,  the  total  gain  being  $3474.30  ? 

16.  The  U.  S.  gold  dollar  (181,  183)  contains  23.22   (25.8 
—A)  grains  of  Pure  gold,  and  the  standard  silver  dollar  371.25 
(412.5— 3*5-)  grains  of  pure  silver.     What  is  the  relative  value  of 
pure  gold  to  pure  silver  ? 

17.  The  assessed  value  of  the  property  of  a  certain  town  is 
$325000,  and  the  total  tax  is  $10238.     How  much  is  the  tax  of 
Mr.  A,  whose  property  is  valued  at  $5700  ? 

18.  A  company  with  a  capital  of  $250000  divides  $8750  among 
its  stockholders.     How  much  will  be  received  by  a  stockholder 
who  owns  36  100-dollar  shares  ? 

19.  If  a  long  ton  of  coal  is  worth  $4.25,  what  is  the  value  of  a 
short  ton  ? 

20.  If  a  farm  valued  at  $4500  is  taxed  $26.24,  what  should  be 
the  tax  on  property  valued  at  $23500  ? 

21.  A  merchant  gains  $625  by  selling  $12000  worth  of  goods  ; 
what  amount  must  he  sell  to  gain  $8000  ? 

22.  How  many  feet  of  boards  will  be  required  for  a  fence  764 
feet  long,  if  888  feet  of  boards  are  required  for  288  feet  ? 

23.  If  one  franc  is  worth  $0.193,   and  one  pound  sterling, 
$4.8665,  what  is  the  value  of  the  pound  sterling  expressed  in 
francs  ? 

24.  If  2175  yards  of  cloth  are  made  from  458  pounds  of  yarn, 
how  many  pounds  of  yarn  would  be  required  to  make  1200  yards 
of  cloth  ? 


216  RATIO     AND     PROPORTION.  [Art.  523. 

25.  If  a  railway  train  goes  412  miles  in   9  lir.  30  min.,  how 
many  hours  would  it  require  to  go  900  miles  ? 

26.  The  railway  fare  from  A  to  B,  a  distance  of  228  miles,  is 
$6.75.     What  should  be  the  fare  from  A  to  C,  a  distance  of  375 
miles  ? 

27.  A  certain  quantity  of  grain  will  last  92  horses  48  days. 
How  long  will  it  last  64  horses  ? 

28.  A  house  and  lot  are  worth  $9600,  and  the  value  of  the  lot 
is  to  the  value  of  the  house  as  5  to  11.     Find  the  value  of  the  lot. 

29.  A  merchant  failing,  owes  $11375,  and  has  property  worth 
$4425.     How  much  will  he  pay  a  creditor  whom  he  owes  $2345  ? 

80.  The  distance  between  two  poles  was  measured  as  48  yards, 
but  the  yard  measure  was  \  of  an  inch  too  short.  What  was  the 
actual  distance  ? 

SI.  From  a  field  of  wheat  containing  375  acres,  4850  bushels 
are  harvested.  How  many  bushels  would  be  harvested  from  a 
field  containing  344  acres  of  similar  wheat  ? 

82.  If  the  tax  on  property,  valued  at  $6500,  is  $144,  what 
should  be  the  tax  on  property  valued  at  $3800  ? 

83.  If  a  Troy  ounce  of   standard   silver   is  worth   85  cents, 
what  is  the  intrinsic  value  of  the  standard  silver  dollar  (112)  ? 

84*  If  the  freight  on  575  pounds  is  $1.84,  what  should  be  the 
freight  on  975  pounds  ? 

85.  If  a  railway  company  charges  $13  for  carrying  one  ton  480 
miles,  what  should  be  the  charge  on  one  ton  for  650  miles  ? 

86.  The  through  rate  from  A  to  C,  a  distance  of  900  miles,  is 
$48  per  car.     What  should  be  the  portion  of  the  A.  &  B.  R.K. 
(425  miles),  and  the  portion  of  the  B.  &  C.  R.R.  (475  miles)? 

87.  If  48  men  can  do  a  certain  piece  of  work  in  60  days,  in 
what  time  can  64  men  do  the  same  work  ? 

38.  If  a  Troy  ounce  (337)  of  silver  is  worth  $1.20,  what  is  the 
value  of  an  Avoirdupois  ounce  ? 

89.  The  ratio  of  the  diameter  of  a  circle  to  its  circumference 
is  1: 3.1416.  What  is  the  circumference  of  a  circle  whose  diameter 
is  475  feet  ? 

40.  If  276  long  tons  of  coal  last  a  manufacturer  21  months, 
how  long  would  276  short  tons  last  him  ? 

41.  A  bankrupt  can  pay  48  cents  on  a  dollar.     If  his  assets 
were  $1887  more,  he  could  pay  65  cents.     Find  his  debts  and  his 
assets. 


Art.  524.]  COMPOUND     PROPORTION.  217 


COMPOUND    PROPORTION. 

524:.  Compound  Proportion  is  an  equality  of  a  compound 
ratio  and  a  simple  ratio,  or  of  two  compound  ratios. 

Th-'  8  ITslVdays  H  *10:  «  »*  4  !  5  M  *  !  »  ""  ^"^ 
proportions. 

Ex.     If  12  men  earn  $60  in  4  days,  how  much  will  10  men 
earn  in  2  days  ? 

°  ANALYSIS. — Since  the  answer  (fourth  term)  is  re- 

4-2  (  ::  60:rr-     <luired  in  dollars,  make  $60  the  third  term  of  the 

proportion.    If  12  men  earn   $60,  16  men  will  earn 


10  *  more;  therefore  make  12  the  first  term   and   16  the 

2  second  term  of  one  ratio.     If  they   earn   $60  in  4 

j*0  days,  in  2  days  they  will  earn  less;  therefore  make  4 

the  first  term  and  2  the  second  term.  The  proportion 
will  then  be  stated  as  in  the  operation.  To  find  the 
fourth  or  unknown  term,  divide  the  continued  product 
of  the  means  by  the  continued  product  of  the  extremes. 

The  quotient  will  be  the  answer.     Apply  cancellation  as  in  the  operation,  by 
placing  the  means  at  the  right  of  a  vertical  line,  and  the  extremes  at  the  left. 

525.  EULE. — Take  for  the  third  term  the  number  that 
may  form  a  ratio  with,  or  is  of  the  same  denomination  as, 
the  answer. 

With  each  pair  of  similar  terms  remaining  form  a 
ratio  as  if  the  result  depended  upon  these  terms  alone. 

Multiply  the  means  together,  and  divide  their  product 
by  the  product  of  the  extremes.  The  quotient  will  be  the 
required  answer. 

NOTE. — Compound  Proportion  is  sometimes  called  the  "Double  Rule  of 
Three." 

EXAM  PLES. 

526.  1.  If  it  costs  $39  to  carpet  a  floor  16  feet  long  and  12 
feet  wide,  what  will  it  cost  to  carpet  a  floor  26  feet  long  and  20 
feet  wide  ? 

2.  If  a  man  can  walk  360  miles  in  12  day3,  walking  8  hours 
each  day,  how  many  hours  a  day  must  he  walk  at  the  same  rate 
to  complete  450  miles  in  20  days  ? 

3.  If  4  men  can  cut  56  acres  of  grass  in  8  days,  how  many 
acres  can  6  men  cut  in  12  days  ? 


218  RATIO     AND     PROPORTION.  [Art.  526. 

4.  If  it  costs  $1728  to  pave  a  street  800  feet  long  and  50  feet 
wide,  what  will  it  cost  to  pave  a  street  1200  feet  long  and  90  feet 
wide  ? 

5.  How  many  hours  a  day  must  42  men  work,  to  do  in  45  days 
what  27  men  can  do  in  28  days  of  10  hours  each  ? 

6.  If  the  gas  for  5  burners,  5  hours  every  evening  for  10  days, 
costs  $2.55,  how  many  burners  may  be  lighted  4  hours  every 
evening  for  15  days  at  a  cost  of  $76.50  ? 

7.  If  22  men  can  dig  a  ditch  4200  feet  long,  5  feet  wide,  and 
3  feet  deep,  in  35  days  of  9  hours  each,  in  how  many  days  of  11 
hours  each  will  252  men  dig  a  ditch  2100  feet  long,  3  feet  wide, 
and  2  feet  deep  ? 

8.  If  3  men  working  11  hours  a  day  can  reap  a  field  of  20 
acres  in  11  days,  in  how  many  days  can  9  men  working  12  hours 
a  day  reap  a  field  360  yards  long  and  320  yards  wide  ? 

9.  A  person  can  read  a  book  containing  220  pages,  each  of 
which  contains  28  lines,  and  each  line  an  average  of  12  words,  in 
5J  hours.     How  long  will  it  take  him  to  read  a  book  containing 
400  pages,  each  of  which  contains  36   lines,  and  each  line  an 
average  of  14  words  ? 

10.  Two  gangs  of  6  and  9  men  are  set  to  reap  two  fields  of  35 
and  45  acres  respectively.     The  first  gang  works  7  hours  a  day, 
and  the  latter  8  hours.     If  the  first  gang  complete  their  work  in 
12  days,  in  how  many  days  will  the  second  gang  complete  theirs  ? 

11.  A  transportation  company  charges  $20  for  carrying  15000 
pounds  400  miles.     How  much  ought  they  to  charge  for  carrying 
60000  pounds  320  miles  ? 

12.  If  6  men  build  a  wall  20  feet  long,  6  feet  high,  and  4  feet 
thick,  in  16  days,  in  what  time  will  24  men  build  a  wall  200  feet 
long,  8  feet  high,  and  6  feet  thick  ? 

IS.  If  the  interest  of  $100  for  1  year  (360  days),  at  6%,  is  $6, 
what  is  the  interest  of  $1200  for  248  days  at  8%  ? 

14.  If  the  freight  of  18  hhd.  of  sugar,  each  weighing  1200  lb., 
for  200  miles,  is  $320,  what  must  be  paid  for  the  freight  of  50  7ihd. 
each  weighing  960  lb.,  fer  420  miles  ? 

15.  If  18  men,  working  10  hours  a  day,  can  hoe  60  acres  in  20 
days,  how  long  will  it  take  50  boys,  working  6  hours  a  day,  to  hoe 
96  acres,  6  men  being  equal  to  10  boys  ? 

16.  If  a  block  of  marble  2x1x3  ft.  weighs  1020  pounds, 
what  is  the  weight  of  a  block  3  x  4  x  8  ft .  ? 


INSURANCE. 


527.  Insurance  is  a  contract  by  which  one  party  (The 
Insurer  or  Underwriter)  engages  for  a  stipulated  consideration 
(The  Premium)  to  make  up  a  loss  which  another  may  sustain. 

Insurance  is  effected  on  property  against  loss  or  damage  by 
fire  and  water,  and  on  lives  of  persons.  (For  Life  Insurance,  see 
Art.  651.) 

Insurance  is  also  effected  against  accidents  to  persons,  the  breakage  of 
plate-glass,  the  loss  of  live  stock,  and  the  dishonesty  of  employees. 

528.  An  Insurance  Company  is  a  company  or  corporation 
which  insures  against  loss  or  damage. 

529.  Insurance   companies   may  be  classified   according  to 
principles  of  organization  as  follows  : — 1,  Stock ;  2,  Mutual ;  3, 
Mixed,  or  Stock  and  Mutual. 

530.  A  Stock  Insurance  Company  is  one  in  which  the 
capital  is  owned  by  individuals,  called  stockholders.     They  alone 
share  the  profits  and  are  liable  for  the  losses. 

The  business  of  a  stock  company  and  also  of  a  mixed  company,  is  managed 
by  directors  chosen  by  the  stockholders.  No  policyholder,  unless  a  stock- 
holder, has  any  voice  in  any  way  in  the  election  of  the  officers,  or  in  the 
management  of  its  business. 

531.  A  Mutual   Insurance   Company  is  one  in  which 
there  are  no  stockholders,  and  the  profits  and  losses  are  shared 
among  those  who  are  insured  (the  policyholders). 

Non-participating  policies,  the  holders  of  which  do  not  share  in  the 
profits  or  losses,  are  issued  by  certain  mutual  and  mixed  companies. 

532.  A  Mixed  Insurance  Company  is  one  which  is  con- 
ducted upon  a  combination  of  the  stock  and  mutual  plan. 

Usually  in  a  mixed  company,  all  profits  above  a  limited  dividend  to  the 
stockholders  are  divided  among  the  participating  policyholders. 


220  INSURANCE.  [Art.  533. 

533.  The   Policy  is  the  contract  between  the  Insurance 
Company  (the  Insurer  or  Underwriter)  and  the  Insured.     It  con- 
tains a  description  of  the  property  insured,  the  amount  of  the 
insurance,  and  the  conditions  under  which  the  policy  is  issued. 

534.  The  Premium  is  the  amount  paid  for  the  insurance. 

1.  Premium  rates  are  expressed  by  giving  the  cost  in  cents  of  $100  insur- 
ance.    The  rate  is  sometimes  expressed  as  a  certain  per  cent,  of  the  amount  of 
the  risk.     Thus,  a  rate  of  75  cents  per  $100  is  equivalent  to  f  %. 

2.  The  premium  rates  depend  upon  the  nature  of  the  risk,  and  the  length 
of  time  for  which  the  policy  is  issued.     The  rate  for  3  years  in  many  Fire 
Insurance  Companies  is  twice  the  rate  for  one  year. 

535.  Short  Rates  are  rates  for  a  term  less  than  a  year. 

If  an  insurance  policy  is  terminated  at  the  request  of  the  policyholder, 
the  company  retains  the  customary  "  short  rates"  for  the  time  the  policy  has 
been  in  force;  if  terminated  at  the  option  of  the  company,  a  ratable  propor- 
tion of  the  premium  is  refunded  for  the  unexpired  term  of  the  policy. 

536.  An  Insurance  Agent  is  a  person  who  represents  an 
insurance  company  or  several  companies,  and  acts  for  them  in 
soliciting  business,  collecting  premiums,  adjusting  losses,  etc. 

537.  An  Insurance  Broker  is  a  person  who  effects  insur- 
ance, for  negotiating  which  he  receives  a  commission  or  brokerage 
from  the  company  taking  the  risk. 

Brokers  are  regarded  as  agents  of  the  insured,  and  not  of  the  insurance 
company. 

538.  The  Surplus  of  an  insurance  company  is  the  excess  of 
the  assets  over  the   liabilities    (including  capital  and   unearned 
premium). 

FIRE     INSURANCE. 

539.  Fire   Insurance   refers  to   insurance  against  loss  or 
damage  by  fire. 

540.  Adjustment  of  Losses. — In  an  ordinary  fire  insur- 
ance policy,  a  person  who  insures  will  be  paid  the  extent  of  his 
loss  up  to  the  amount  of  his  insurance  ;  but  in  policies  contain- 
ing the    "average  clause/'  the  payment  is  such  proportion   of 
the  loss  as  the  amount  of  the  insurance  bears  to  the  total  value 
of  the  property. 


Art.  540.]  MARINE     INSURANCE.  221 

1.  The  following  is    the  usual  form  of  the    "average  clause"   above 
referred  to  :    "It  is  a  condition  of  this  insurance,  that  if  the  whole  value  of 
the  above  described  property,  contained  in  any  or  all  of  the  above  mentioned 
buildings  and  premises,  shall  exceed  the  whole  amount  of  insurance  thereon, 
then,  in  case  of  loss  or  damage  by  fire,  this  policy  shall  contribute  to  the 
payment  of  said  loss  or  damage  in  the  proportion  only  that  the  whole  amount 
of  insurance  on  said  property  shall  bear  to  the  whole  value  of  said  property, 
in  all  of  said  buildings,  at  the  time  said  loss  or  damage  may  occur." 

2.  Under  a  policy  containing  the  "  average  clause,"  a  person  who  insures 
$5000  on  property  worth  $10000,  would  receive  only  $2500  in  case  of  an  actual 
loss  of  $5000;  $1500  in  a  loss  of  $3000. 

3.  Insurance  companies  usually  reserve  the  privilege  of  replacing  or 
repairing  the  damaged  premises. 


MARINE     INSURANCE. 

541.  Marine  Insurance  refers  to  insurance  of  vessels  and 
their  cargoes  against  the  dangers  of  navigation. 

1.  Inland  and    Transit    Insurance   refer    to  insurance  of  merchandise 
while  being  transported  from  place  to  place  either  by  rail  or  water  routes, 
or  both. 

2.  Policies  on  cargoes  are  issued  for  a  certain  voyage,  or  from  port  to 
port,  and  on  vessels  for  a  specified  time  or  for  a  certain  voyage.  , 

3.  The  particular  average  clause  is  the  clause  which  exempts  the  insur- 
ance company  from  the  payment  of  any  partial  loss  or  particular  average, 
unless  it  exceeds  a  certain   per  cent,  of  the  value  of  the  property.     The 
particular  average  clause  is  sometimes  applied  to  the  value  of  each  parcel  or 
series  of  parcels,  according  to  invoice  numbers. 

4.  Insurance  Certificates,  showing  that  certain  property  has  been  insured, 
and  stating  the  amount  of  the  insurance  and  the  name  of  the  party  abroad 
who  is  authorized  to  make  the  settlement,  are  issued  by  marine  companies. 
They  are  negotiable,  and  are  usually  sent  to  the  consignee  of  the  merchandise 
to  make  the  loss  payable  at  the  port  of  destination,  and  to  otherwise  facilitate 
the  adjustment  of  the  insurance  in  case  of  loss. 

542.  Adjustment   of  Losses. — In  marine  insurance,  in 
case  of  loss  or  damage,  the  insurance  company  contributes  such 
proportion  of  the  loss  as  the  amount  of  the  insurance  bears  to  the 
total  value  of  the  property. 

1.  The  adjustment  of  marine  losses  is  on  the  same  principle  as  the  adjust- 
ment of  fire  policies  containing  the  " average  clause"  (54O,  1). 

2.  In  the  adjustment  of  marine  losses,  the  pound  sterling  is  usually 
estimated  at  $4.95. 


222  INSURANCE.  [Art.  543. 

543.  An  Open  Policy  is  one  upon  which  additional  insur- 
ances may  be  entered  at  different  times.     It  covers  merchandise 
which  may  be  shipped  on  "  Vessel  or  Vessels  "  from  "  Ports  and 
Places"  to  "Ports  and  Places,"  for  amounts  "as  endorsed"  and 
at  rates  "as  agreed." 

1.  The  date  of  the  shipment,  name   of  vessel,  ports  of  shipment  and 
destination,  the  amount  of  the  insurance,  rate,  premium,  and  a  description 
of  the  property  are  entered  on  the  policy  or  in  a  pass-book,  which  is  regarded 
as  part  of  the  policy.     (See  Ex.  27,  Art.  544.) 

2.  Open  policies  with  pass-books  attached  and    insuring  merchandise 
against  loss  or  damage  by  fire,  are  issued  by  fire  insurance  companies. 

3.  Open  policies,  which  cover  all  risks  whether  accepted  and  endorsed  on 
the  policy  or  not,  are  issued  to  merchants  who  are  receiving  merchandise  from 
foreign  countries,  and  who  do  not  always  have  a  definite  knowledge  of  the 
time  of  shipment.    Such  policies  usually  contain  the  following  clause  :  "  The 
company  are  to  be  entitled  to  premiums  at  their  usual  rates  on  all  shipments 
reported  or  not.     It  is  warranted  by  the  assured  to  report  every  shipment  on 
the  day  of  receiving  advice  thereof,  or  as  soon  thereafter  as  practicable,  when 
the  rate  of  premium  shall  be  fixed  by  the  President  of  the  Company." 

The  above  policies  cover  the  invoice  cost  and  10^  additional  until  the 
amount  of  the  risk  is  endorsed  on  the  policy  or  pass-book. 

4.  Open  policies  are  sometimes  issued  which  cover  only  such  risks  as 
may  be  accepted  and  endorsed  on  the  policy  by  the  company. 

EXAMPLES. 

544.  1.  A  building  was  insured  for  $2500  in  one  company  at 
1J%,  and  for  $5000  in  another  company  at  125  cents.     What  was 
the  total  premium  paid  ? 

2.  A  cargo  of  goods  was  insured  for  $9000  at  \%.     What  was 
the  cost  of  the  insurance,  $1.25  being  charged  for  the  policy  ? 

3.  What  is  the  total  premium  of  the  following  insurances  : 
$5000  at  \\%,  $7000  at  45^,  $2000  at  5$,  $3500  at  45^,  $2000  at 
70$*,  $4000  at  \\%,  $2000  at  60^,  $4500  at  25^,  and  $3600  at  125$*? 

4.  $20  was  paid  for  an  insurance  of   $2500 ;  what  was  the 
premium  rate  ? 

5.  $25.20  was  paid  for  an  insurance  at  the  rate  of  70^  per 
$100.     What  was  the  amount  of  the  risk  ? 

6.  A  factory  was  insured  for  $7500  for  1  year  at  2£%,  stock  for 
$2500  at  %\%,  and  raw  material  for  $2500  at  1J$.     What  was  the 
total  premium  ? 

7.  What  is  the  cost  of  insuring  a  house  for  $5000  at  the  rate 
of  45^  per  $100  ? 


Art.  544.]  INSURANCE.  223 

8.  A  cargo   of  merchandise  was  insured  for  $6500  at  |%, 
including  the  risk  of  fire  while  on  wharf   awaiting   shipment. 
What  was  the  premium  ? 

9.  A  building  was  insured  Jan.  1, 1880,  for  $2000,  for  7  years,  at 
5% ;  what  was  the  value  of  the  unearned  premium,  Jan.  1,  1882  ? 

10.  A  shipment  of  goods  was  insured  in  the  Pacific  Mutual 
Insurance  Co.  for  $9600  at  75^  less  20%  in  lieu  of  scrip  and  inter- 
est.    What  was  the  net  cost  of  the  insurance  ? 

11.  A  house  was  insured   for   $5000   for  4  years  at  60^   per 
annum.     The  house  was  destroyed  by  fire.     What  was  the  actual 
loss  of  the  company,  making  no  allowance  for  interest  ? 

12.  Find  the  cost  of  insuring  a  house  for  3  years  for  $4000  at 
60^,   and   the   furniture   for   $1200   at   80^,    less   15%   on    both 
premiums. 

13.  A  cargo  of  hides  having  increased  in  value  since  the  insur- 
ance was  effected,  the  anticipated  profits  were  insured  for  $3000  at 
1J%  less  20%.     What  was  the  premium  ? 

14.  A  factory  (worth  $3000)  and  its  contents  are  insured  for 
$10000  as  follows  :  $2000  on  building,  $3000  on  machinery  (worth 
$5000),  and   $5000   on  stock  (worth   $8000).     The  building   is 
damaged  by  fire  to  the  amount  of  $1000,  the  machinery  $4000, 
and  stock  is  a  total  loss.     How  much  is  the  claim  against  the 
insurance  company  ? 

15.  A  cargo  of  goods  valued  at  $20000  was  insured  for  $12000. 
If  the  goods  were  damaged  to  the  amount  of  $15000,  how  much 
of  the  loss  would  be  paid  by  the  insurance  company  ?     (542.) 

16.  A  building  is  insured  in  several  companies  for  $60000,  and 
is  damaged  by  fire  to  the  extent  of  $24000.     What  per  cent,  of  its 
risk  is  paid  by  each  company  ? 

17.  A  stock  of  goods  was  insured,  May  1,  for  1  year,  for  $6000, 
at  90^.     The  policy  was  cancelled  Nov.  1,  at  the  request  of  the 
insured.     How  much  was  the  return  premium,  the  short  rate  for 
6  months  being  63^  ?    How  much  would  have  been  returned  by 
the  company,  if  the  policy  had  been  cancelled  at  its  request  ? 

18.  A  quantity  of  merchandise  valued  at  $6000  is  insured  for 
$5000.     It  is  damaged  by  fire  to  the  amount  of  $1728.     How 
much  of  the  loss  is  paid  by  the  insurance  company,  the  policy 
containing  the  "average  clause"  ?   (54O.) 

19.  What  was  paid  for  insuring  a  cargo  of  merchandise  for 
$8750  at  |%  less  20%  ? 


224: 


IlfS  URANCE. 


[Art.  544. 


20.  A  factory  and  its  contents  are  insured  for  '$5000  in  com- 
pany M,  $5000  in  N,  $5000  in  0,  $4000  in  P,  and  $2500  in  each 
of  the  following  companies :    Q,  K,  S,  T,  U,  V,  W,  X,  Y,  and  Z. 
What  was  the  total  premium,  the  rate  being  %%  less  10$  ? 

21.  The   above   insurance   covered   the    following    property : 
$4000  on  building  marked  A  on  plan,  $4000  on  B,  $5000  on  C, 
$500  on  D,  $500  on  E,  $3500  on  stock  and  materials  in  building 
marked  A  on  plan,  $8000  on  machinery,  etc.,  in  A,  $11500  on 
stock  and  materials  in  B  and  C,  $4000  on  machinery,  etc.,  in  B 
and  C,  $2500  on  horses  in  D,  $500  on  harness,  hay,  feed,  etc., 
in  D.     Suppose  building  A  and  its  contents  were  totally  destroyed 
by  fire,  what  would  be  the  loss  of  company  M  ?     Of  P  ?     Of  T  ? 

NOTE.— The  above  insurance  was  divided  pro  rata  among  the  several 
companies,  each  policy  designating  the  exact  amount  on  each  building,  etc. 

22.  In  the  above  example,  what  is  the  amount  of  the  risk  of 
company  M  on  the  building  marked  A  on  plan  ?     On  C  ? 

28.  The  net  invoice  value  of  a  quantity  of  goods  is  $6325,  an<i 
the  insured  value  $6500.  The  insured  value  is  what  per  cent, 
greater  than  the  invoice  value? 

24..  A  quantity  of  merchandise  valued  at  $9035,  is  insured  for 
$9000.  What  is  the  insurance  on  part  of  the  same,  the  estimated 
value  being  $2638  ? 

25.  If  500  packages  of  merchandise  arc  insured  for  $2627.78, 
what  is  the  insurance  on  60  packages  ? 

26.  The  estimated  sound  value  of  a  quantity  of  merchandise, 
damaged  at  sea,  was  $328.55,  and  the  proceeds  when  sold  at  auc- 
tion, $299.35.     How  much  of  the  loss  was  shared  by  the  Insurance 
Co.,  the  insurance  having  been  $315.33  ? 

27.  Make  the  extensions  of  the  following  "open  policy"  and 
find  the  total  amount. 


Date. 

Sept.  '  2 

"      7 
"    16 
"    17 

Name  of 
vessel. 

From. 

To. 

On. 

Amount 
insured. 

Rate. 

Hi 

Othello. 
Algeria. 
Germanic. 
Rialto. 

N.Y.  via  Hull. 
New  York. 
New  York. 
N.Y.  via  Hull. 

Stockholm. 
Liverpool. 
Liverpool. 
Christiama. 

50Ba.Mdse. 

68  " 
92  " 
6  "        " 

5100 
6675 
13500 
600 

H 
1 
1 

1 

**  ** 
»*_** 

"    23 

Otranto. 

N.Y,  via  Hull. 

Orebro. 

30  "        " 

2700 

T.Aoa  < 

n 
>.ftct 

**.** 

^**«-  ** 
**'** 

#**,** 

EXCHANGE. 


545.  Exchange  is  the  system  by  which  merchants  in  distant 
places  discharge  their  debts  to  each  other  without  the  transmission 
of  money. 

Suppose,  for  example,  A  of  New  York  owes  B  of  Chicago  $1000  for  grain, 
and  C  of  Chicago  owes  D  of  New  York  $1000  for  dry  goods.  The  two  debts 
may  be  discharged  by  means  of  one  draft  or  bill  of  exchange  without  the 
transmission  of  money.  Thus,  B  of  Chicago  draws  on  A  of  New  York  for 
$1000,  and  sells  the  draft  to  C  of  Chicago,  who  remits  it  to  D  of  New  York. 
D  of  New  York  presents  the  draft  to  A  of  New  York  for  acceptance  or  pay- 
ment, and  thus  both  debts  are  cancelled.  There  is  in  effect  a  setting-off  or 
exchange  of  one  debt  for  the  other. 

The  business  of  exchange  is  usually  conducted  through  the  medium  of 
banks  and  bankers,  who  buy  commercial  bills  and  transmit  them  for  credit  to 
the  places  on  which  they  are  drawn.  They  also  sell  their  own  drafts  on  their 
correspondents  in  any  amounts  demanded. 

The  greater  part  of  the  exchange  in  the  United  States  is  effected  through 
the  banks  of  New  York,  Boston,  Philadelphia,  Chicago,  St.  Louis,  Baltimore, 
and  San  Francisco.  The  financial  centres  of  Europe  are  London,  Paris, 
Antwerp,  Geneva,  Amsterdam,  Hamburg,  Frankfort,  Bremen,  Berlin,  and 
Vienna. 

546.  A  Bill  of  Exchange,  or  Draft,  is  an  order  or  request 
addressed  by  one  person  (the  Drawer)  to  another  (the  Drawee), 
directing  the  payment  of  a  specified  sum  of  money  to  a  third  per- 
son (the  Payee)  or  to  his  order.     It  is  issued  at  one  place  and 
payable  at  another.     (See  Art.  495,  5-6.) 

For  brevity,  bills  of  exchange  are  frequently  called  "exchange." 
According  to  the  laws  of  most  States,  drafts  drawn  in  one  State  and  pay- 
able in  another,  are  termed  foreign  bills  of  exchange.     For  the  purposes  of 
this  book,  the  term  "  domestic  exchange"  will  be  applied  to  bills  drawn  and 
payable  in  the  United  States. 

547.  Bills  of  exchange  are  of  two  kinds,  Inland  or  Domestic, 
and  Foreign. 

548.  A  Domestic  or  Inland  Bill  of  Exchange  is  ona 
which  is  payable  in  the  same  country  in  which  it  is  drawn. 


226  EXCHANGE.  [Art.  549. 

549.  A  Foreign  Bill  of  Exchange  is  one  which  is  payable 
in  a  different  country  from  the  one  in  which  it  is  drawn  ;  as  a 
draft  drawn  in  the  United  States  and  payable  in  England. 

05 O.  When  drafts  sell  for  more  than  their  face  value,  exchange 
is  above  par  or  at  a  premium  ;  when  for  less  than  their  face,  below 
par  or  at  a  discount. 

When  Chicago  owes  New  York  the  same  amount  that  New  York  owes 
Chicago,  exchange  will  be  at  par  ;  that  is,  drafts  will  sell  at  their  face  value. 
When  Chicago  owes  New  York  more  than  New  York  owes  Chicago,  drafts  on 
New  York  will  sell  at  a  premium  ;  there  will  be  more  buyers  of  exchange 
than  sellers,  and  drafts  will  sell  for  more  than  their  face  value.  When  Chicago 
owes  New  York  less  than  New  York  owes  Chicago,  the  demand  in  Chicago 
for  drafts  on  New  York  will  be  less  than  the  supply,  and  drafts  will  sell  for 
less  than  their  face  value,  or  at  a  discount. 


DOMESTIC     EXCHANGE. 

551o  Domestic  or  Inland  Exchange  relates  to  drafts  drawn 
at  one  place  on  another  in  the  same  country. 

552.  The  domestic  exchanges  on  New  York  at  the  places 
named  were  quoted  as  follows,  May  7,  1881  :  Savannah,  -J-  @  f 
premium;  Charleston,  -J  @  J-  premium;  New  Orleans,  $1.50  @ 
$2.50  premium;  St.  Louis,  25  cents  premium  ;  Chicago,  50  @  75 
cents  premium  ;  and  Boston,  25  cents  discount. 

1.  At  Savannah  and  Charleston  the  rates  per  cent,  of  the  premium  or 
discount  are  given.     Thus,  when  exchange  is  quoted  at  -|-  premium,  a  draft  of 
$100  may  be  purchased  for  $100}  ($100.25). 

2.  At  New  Orleans,  St.  Louis,  Chicago,  and  Boston,  the  premium  or  dis- 
count per  $1000  is  given.     Thus,  a  draft  of  $1000  at  $2.50  premium  may  be 
purchased  for   $1002.50.      $2.50  per  $1000  premium   is   equivalent  to  \% 
premium. 

3.  The  selling  rates  are  about  \%   ($1.25)  higher  than  the  buying  rates, 
and  bankers'  exchange  is  usually  higher  than  commercial. 

4.  The  rate  of  domestic  exchange  is  limited  by  the  cost  of  shipping  gold 
or  currency  by  express,  and  the  premium  or  discount  will  not  exceed  this  cost. 
Thus,  if  a  merchant  in  Chicago  is  charged  a  premium  of  $10  for  a  draft  of 
$10000,  and  lie  can  send  the  currency  by  express  for  $7.50,  it  will  be  to  his 
advantage  to  remit  by  the  latter  method. 

The  following  appeared  in  a  New  York  financial  paper,  May  8,  1881,  the 
date  of  the  above  quotations  :  — "  The  domestic  exchanges  at  the  West  are 
sufficiently  high  to  permit  of  a  movement  of  funds  Eastward,  but  at  the  East, 


Art.  552.]  EXCHANGE.  227 

New  York  funds  are  still  at  a  discount  and  some  shipments  of  gold  and 
currency  continue  to  be  made  to  the  Eastern  cities." 

5.  The  preceding  quotations  refer  to  sight  exchange.  Time  drafts  are  dis- 
counted in  the  same  manner  as  promissory  notes.  In  certain  cases  bankers  in 
discounting  notes  and  drafts  payable  in  distant  places,  charge  interest  for  the 
time  required  for  the  return  of  tho  money  when  the  note  or  draft  is  paid  ;  and 
in  the  case  of  drafts  drawn  a  certain  number  of  days  after  sight,  bankers 
sometimes  charge  interest  for  the  time  required  for  the  acceptance  of  the 
drafts.  Thus,  if  a  draft  was  drawn  in  New  York  on  St.  Louis  and  payable 
60  days  after  sight,  it  would  require,  in  the  ordinary  course  of  the  mails,  3 
days  for  the  acceptance  of  the  draft.  The  draft  would  be  paid  in  63  days 
(including  the  days  of  grace),  and  3  days  would  elapse  before  the  money 
would  be  returned  to  New  York.  The  banker  would  be  justified  in  charging 
interest  for  69  days,  the  interval  between  the  day  he  advanced  the  money  in 
New  York,  and  the  day  it  was  returned  to  him  again.  If  the  draft  was  drawn 
on  San  Francisco,  fully  19  days  (8  days  for  the  acceptance,  3  days  of  grace, 
and  8  days  for  the  return  of  the  money)  would  be  added  to  the  time  of  the 
draft.  Between  New  York  and  San  Francisco  and  other  distant  places,  money 
is  frequently  transferred  by  telegraph.  (See  Art.- 499,  3.) 

EXAMPLES. 

553.  1.  What  is  the  value  in  Savannah  of  a  draft  on  New 
York  for  $8750  at  \%  premium  ? 

2.  Find  the  cost  in  New  Orleans  of  a  draft  on  New  York  for 
$8375  at  $2.50  premium. 

Find  the  Value  of  the  following  drafts  : 

Face.  Exchange.  Face.  Exchange. 

5.15000,  \%  premium.  8.  $4287.75,         15?  discount. 

4.  $4375,  \%  discount.  9.  83416.33,         25?  premium. 

6.  $8417,  \%  premium.  10.  $2825.49,    $1.25    discount. 

6.  $9873,  \%  premium.  11.  $9873.62,    $2.50   premium. 

7.  $5284,  £%  discount.  .£2.  $8412.75,         75^  discount. 
13.  A  of  Chicago   buys  cattle   for   B   of-  New  York  to  the 

amount  of  $9858.07.  How  large  a  draft  should  be  drawn  on  B, 
so  that  when  sold  at  a  discount  of  5W  (-£$%},  the  proceeds  would 
be  sufficient  to  pay  the  bill  ? 

NOTE. — To  find  the  face  of  a  draft,  instead  of  dividing  the  value  of  the 
draft  by  the  rate  of  exchange  (in  the  above  example,  .99^f  or  .9995),  business 
men  and  bankers  calculate  the  premium  or  discount  on  the  value  of  the  draft, 
and  subtract  or  add  it  to  the  value  as  the  case  requires.  Thus,  in  the  above 
example,  the  discount  would  be  \  of  -fa%  of  $9858.07,  or  $4.93,  which  added 
to  the  given  proceeds  would  produce  the  face  $9863.  This  method  produces 
too  small  a  result  in  all  cases,  the  error  being  equivalent  to  the  percentage  of 
the  premium  or  discount.  In  this  example  the  error  is  less  than  \  cent. 


EXCHANGE.  [Art.  553. 

For  ordinary  examples  in  business,  the  foregoing  method  is  sufficiently- 
accurate.  At  \  % ,  or  $5.00  (a  very  high  rate  for  domestic  exchange)  on  a  draft 
whose  value  is  $10000,  the  error  would  be  only  25  cents.  If  greater  accuracy 
is  required,  the  necessary  correction  can  be  made  by  adding  the  percentage 
of  the  premium  or  discount.  Thus,  if  the  value  of  the  draft  is  $10000,  and 
exchange  is  \%  discount,  the  face  would  be  $10000  +  $50  (\%  of  $10000) 
+  $0.25  (\%  of  $50)  =  $10050.25.  If  at  \%  premium,  the  face  would  be 
$10000  -  $50  +  $0.25  =  $9950.25. 

By  the  above  method,  find  the  face  of  the  following  drafts : 

Value.  Exchange.  Value.  Exchange. 

14.  $1876.16,  J#  premium.  19.  $7375,  25^  premium. 

15.  $2437.75,  J^  discount  20.  $9218,  50^  discount. 

16.  $3342.38,  \%  discount.  21.  $6438,  $1.00    premium. 

17.  $2238.42,  J$  premium.  22.  $9243,  $1.25    premium. 

18.  $8175.50,  \%  premium.  28.  $5280.  75^  discount. 

24.  A  of  New  Orleans  being  indebted  to  B  of  New  York 
$9316.75,  forwards  to  him  a  check  on  a  New  Orleans  bank  for 
that  amount,  to  cash  which  B  is  obliged  to  allow  a  discount  of 
\%.     How  much  does  A  still  owe  B,  and  for  what  amount  should 
theicheck  have  been  drawn  to  net  B  the  amount  due  ? 

25.  What  is  the  value  of  a  draft  on  New  York  for  $3000, 
payable  in  60  days  (63  days)  after  date  (494,  7),  exchange  being 
J$  premium,  and  interest  6%? 

NOTE. — From  the  face  of  the  draft,  subtract  the  interest,  and  to  the 
result  add  the  exchange. 

26.  Find  the  proceeds  of  a  draft  drawn  at  Chicago  on  New 
York  for  $12000,  and  payable  90  days  after  sight,  exchange  50^ 
discount,  interest  5$,   and  allowing  3  days   additional  for  the 
acceptance  of  the  draft. 

27.  A  banker  in  New  York  discounts  a  draft  for  $8000,  pay- 
able in  San  Francisco  60  days  after  sight  ;  what  would  be  the 
proceeds,  exchange  being  \%  discount,  interest  6%,  and  allowing 
8   days  for  the   acceptance   and   8  days  for  the  return   of  the 
money  ? 

28.  A  merchant  paid  $6920.64  in  Charleston  for  a  sight  draft 
of  $6912  ;  what  was  the  rate  of  exchange  ? 

29.  A  commission  merchant  sold  13475  pounds  of  leather  at 
26}  cents  a  pound.     If  his  commission  is  5%,  and  exchange  \% 
premium,  how  large  a  draft  can  he  buy  to  remit  to  the  consignor  ? 

SO.  How  large  a  60-days'  draft  must  I  draw,  so  that  when  sold 
it  will  produce  $10000,  exchange  \%  discount,  interest 


Art.  554.]  FOREIGN    EXCHANGE.  229 


FOREIGN     EXCHANGE. 

554.  Foreign  Exchange  relates  to  drafts  or  bills  of  exchange 
drawn  in  one  country  and  payable  in  another. 

555.  To  secure  safety  and  speed  in  the  transmission  of  foreign 
bills  of  exchange,   they  are  drawn  in  sets  of  two  or  three  of  the 
same  tenor  and  date.     The  separate  bills  are  sent  by  different 
steamers,  and  when  any  one  of  them  is  paid,  the  others  become 
void.     Some  merchants  send  only  the  first  and  second,  and  pre- 
serve the  third. 

SET     OF     EXCHANGE. 

(I-) 

EXCHANGE  FOR  £1000.  NEW  YORK,  May  16,  1889. 

Sixty  days  after  sight  of  this  FIRST  of  Exchange  (Second  and 
Third  unpaid),  pay  to  the  order  of  A.  T.  STEWART  &  Co.,  One 
Thousand  Pounds  Sterling,  value  received,  and  charge  the  same 
to  account  of 

No.  1738.  BROWN  BROTHERS  &  Co. 

To  BROWN,  SHIPLEY  &  Co.,  | 
London,  England.      j 

(2.) 

EXCHANGE  FOR  £1000.  NEW  YORK,  May  16,  1889. 

Sixty  days  after  sight  of  this  SECOND  of  Exchange  (First  and 
Third  unpaid),  pay  to  the  order  of  A.  T.  STEWART  &  Co.,  One 
Thousand  Pounds  Sterling,  value  received,  and  charge  the  same 
to  account  of 

No.  1738.  BROWN  BROTHERS  &  Co. 

To  BROWN,  SHIPLEY  &  Co.,  \ 
London,  England.      J 

(3.) 

EXCHANGE  FOR  £1000.  NEW  YORK,  May  16,  1889. 

Sixty  days  after  sight  of  this  THIRD  of  Exchange  (First  and 
Second  unpaid),  pay  to  the  order  of  A.  T.  STEWART  &  Co.,  One 
Thousand  Pounds  Sterling,  value  received,  and  charge  the  same 
to  account  of 

No.  1738.  BROWN  BROTHERS  &  Co. 

To  BROWN,  SHIPLEY  &  Co.,  ] 
London,  England.      J 


230  EXCHANGE.  [Art.  555. 

Foreign  bills  of  exchange  are  usually  drawn  in  the  moneys  of  account  of 
the  countries  in  which  they  are  payable.  Thus,  drafts  on  England  are  usually 
drawn  in  pounds,  shillings,  and  pence;  on  France,  Belgium,  and  Switzerland, 
in  francs;  on  Germany,  in  marks;  on  the  Netherlands  (Holland),  in  guilders. 

Foreign  bills  of  exchange  are  usually  drawn  at  sight  (3  days)  or  at  sixty 
(63  days)  days'  sight.  Sight  drafts  are  frequently  called  "  short  "  exchange, 
and  60  day  drafts,  "long"  exchange.  "Long"  exchange  is  sold  at  a  rate 
below  that  for  "  short  "  exchange,  sufficient  to  equalize  the  difference  in 
interest  between  the  dates  of  maturity  of  the  two  classes  of  bills. 

556.  A  Letter  of  Credit   is  an   instrument   issued  by  a- 
banker  and  addressed  to  bankers  generally,  by  which  the  holder 
may  draw  funds  at  different  places  and  in  amounts  to  suit  his 
convenience,,  the  total  amount  drawn  not  exceeding  the  limit  of 
the  letter  of  credit. 

A  bill  of  exchange  is  payable  at  a  certain  place,  at  a  certain  fixed  time, 
and  for  a  certain  amount,  while  a  letter  of  credit  is  payable  at  different  places, 
at  different  times,  and  in  different  amounts. 

A  person  who  intends  to  travel  in  foreign  countries,  may  procure  a  letter 
of  credit  by  depositing  either  cash  or  securities  with  a  foreign  exchange 
banker  for  the  amount  of  the  letter.  When  the  American  banker  is  notified 
of  the  payment  of  the  traveler's  drafts  in  London,  he  debits  the  account  erf 
the  holder  of  the  letter  of  credit  with  the  amount  drawn  and  the  charges,  at 
the  current  rate  of  exchange.  A  small  rate  of  interest  is  sometimes  allowed 
on  the  account,  and  a  settlement  is  made  on  the  return  of  the  traveler. 

557.  The  Intrinsic  Par  of  Exchange  is  the  value  of  the 
monetary  unit  of  one  country  expressed  in  that  of  another,  and  is 
based  on  the  comparative  fineness  and  weight  of  the  coins,  as 
determined  by  assay.     (See  Art.  566.) 

558.  The  Commercial  Par  of  Exchange  is  the  market 
value  in  one  country  of  the  coins  of  another. 

559.  The  Commercial  Bate  of  Exchange  is  the  market 
or  buying  and  selling  value  in  one  country  of  the  drafts  on  another. 

1.  In  giving  quotations  of  foreign  exchange,  no  reference  is  made  to  the 
par  value,  the  quotations  being  given  by  means  of  equivalents. 

2.  Premium  or  discount  for  exchange  cannot  long  exceed  the  transporta- 
tion charges  and  insurance  of  shipping  coin;  for,  if  a  merchant  can  ship  gold 
cheaper  than  he  can  buy  a  bill  of  exchange,  he  will  choose  the  former  method 
of  paying  his  indebtedness.     When  sight  exchange  is  4.84,  gold  can  be  im- 
ported at  a  small  profit;  and  when  sight  exchange  is  4.89£,  gold  can   be 
exported  at  a  profit. 


Art.  559.]  FOREIGN    EXCHANGE.  231 

3.  When  exchange  is  above  par,  we  are  exporters  of  gold ;  when  below 
par,  we  are  importers  of  gold. 

560.  Exchange  on  England  (Sterling  exchange)  is  quoted  by 
giving  the  value  of  £1  in  dollars  and  cents. 

Thus,  when  exchange  is  4.84,  a  draft  of  £1  will  cost  $4.84;  of  £100,  $484. 
The  intrinsic  par  value  of  £1  is  $4.8665  (566). 

561.  Exchange  on  Franqe,  Belgium,  and  Switzerland  is  quoted 
by  giving  the  value  of  $1  in  francs  and  centimes  (hundredths  of  a 
franc). 

Thus,  when  exchange  is  5.27^,  $1  will  buy  a  bill  of  5  francs  and  27-|- 
centimes;  a  draft  of  1000  francs  will  cost  $189.57  (1000  -H  5.27|).  The 
intrinsic  par  value  of  1  franc  is  19^-  cents  (566) ;  of  the  equivalent  exchange, 
5.18J  (1.00  -4-  .193). 

In  French,  Belgian,  and  Swiss  exchange,  the  higher  the  apparent  rate, 
the  less  the  value  of  the  draft.  Thus,  when  exchange  is  5.13,  a  draft  of  1000 
francs  is  worth  $194.93,  and  each  franc  is  worth  19^  cents.  When  exchange 
is  5.26f,  the  same  draft  would  be  worth  $189.98,  and  each  franc  19  cents. 

562.  Exchange   on  Amsterdam  (Netherlands)  is  quoted  by 
giving  the  value  of  one  guilder  (gulden)  or  florin  in  U.  S.  cents. 

The  intrinsic  par  value  of  1  guilder  is  40T2TI  cents  (566). 

563.  Exchange  on  Germany  is  quoted  by  giving  the  value  of 
4  reichsmarks  in  cents. 

The  intrinsic  par  value  of  1  mark  is  23T%  cents  (566) ;  of  4  marks  95^ 
cents. 

564.  Documentary  Exchange  is  a  bill  drawn  by  a  shipper 
upon  his  consignee  against  merchandise  shipped,  accompanied  by 
the  letter  of  hypothecation,  the  bill  of  lading  "  to  order,"  and  the 
insurance  certificates  covering  the  property  against  which  the  bill 
is  drawn. 

565.  Exchange  on  London  in  the  countries  named,  and  at 
London  on  the  same  countries,  is  quoted  as  follows : 

United  States,  by  giving  the  value  of  £1  in  dollars  and  cents. 
France  and  Belgium,  by  giving  the  value  of  £1  in  francs  and  centimes. 
Germany,  by  giving  the  value  of  £1  in  marks  and  pfenniges. 
Austria,  by  giving  the  value  of  £1  in  florins  and  kreutzers. 
Netherlands,  by  giving  the  value  of  £1  in  guilders  and  cents. 
India,  by  giving  the  value  of  1  rupee  in  shillings  and  pence. 


232 


EXCHANGE. 


[Art.  566. 


566.   FOREIGN  MONEYS  OF  ACCOUNT. 


Country. 

Standard. 

Monetary  Unit. 

Value  in 
U  S.  Gold. 

Argentine  Republic. 
Austria, 

Gold  and  silver. 
Silver 

Peso  of  100  centavos  .  .  . 
Florin  of  100  kreutzers  . 

.96,5 
.359 

Gold  and  silver. 

"Franc  of  100  centimes.  . 

.19,3 

Bolivia  

Silver  

bBoliviano,100  centavos 

.72,7 

Brazil 

Gold  

Milreis  of  1000  reis.  . 

.54,6 

British  America/ 

Gold 

Dollar  of  100  cents      . 

$1.00 

Chili  

Gold  and  silver. 

Peso  of  100  centavos  .  .  . 

.91,2 

Cuba  

Gold  and  silver. 

Peso  of  100  centavos..  . 

.93,2 

Denmark  .... 

Gold  

c  Crown  of  100  ore  

.26,8 

Ecuador. 

Silver  

b  Sucre  of  100  centavos  . 

.72,7 

Effvpt 

Gold  

Pound  of  100  piasters 

4.943 

France  

Gold  and  silver. 

'Franc  of  100  centimes.  . 

.19,3 

German  Empire 

Gold.     . 

Mark  of  100  pfennige 

238 

Great  Britain 

Gold          .      . 

Pound  of  20  shillings 

4866i 

Greece    

Gold  and  silver. 

•Drachma  of  100  lepta 

.19,3 

Hayti  

Gold  and  silver. 

d  Gourde  of  100  centavos 

.96,5 

India                       . 

Silver  

Rupee  of  16  annas6 

.34,6 

Italy 

Gold  and  silver. 

ttLira  of  100  centesimi 

.193 

Gold  and  silver. 

YenoflOOsen-SGold" 

.99,7 

Liberia               . 

Gold  

(  Silver. 
Dollar  of  100  cents  .  . 

.78,4 
1  00 

Mexico     .  .           ... 

Silver  

Dollar  of  100  centavos. 

79 

Netherlands 

Gold  and  silver. 

Florin  of  100  cents 

402 

Norway      . 

Gold.   ... 

c  Crown  of  100  ore 

26  8 

Peru 

Silver  

bSol  of  100  centavos 

72  7 

Portugal  

Gold  

Milreis  of  100  reis 

108 

Russia  

Silver  

Rouble  of  100  copecks  . 

582 

Spain  

Gold  and  silver. 

"Peseta  of  100  centimes 

193 

Sweden 

Gold  

c  Crown  of  100  ore 

268 

Switzerland 

Gold  and  silver. 

*Franc  of  100  centimes 

193 

Tripoli 

Silver            

Mahbub  of  20  piasters 

656 

Turkey  

Gold  

Piaster  of  40  paras.   . 

044 

U.  S.  of  Colombia.  . 

Silver  

bPeso  of  100  centavos 

72  7 

Venezuela  

Gold  and  silver. 

*Bolivar  of  100  centavos 

193 

The  above  rates,  proclaimed  by  the  Secretary  of  the  Treasury,  Jan.  1,  1887,  are  used  in 
estimating,  for  Custom  House  purposes,  the  values  of  all  foreign  merchandise  made  out  in 
any  of  said  currencies. 

(a)  The  franc  of  France,  Belgium,  and  Switzerland,  the  peseta  of  Spain,  the  drachma  of 
Greece,  the  lira  of  Italy,  and  the  bolivar  of  Venezuela  have  the  same  value. 

(b)  The  sucre  of  Ecuador,  the  peso  of  United  States  of  Colombia,  the  boliviano  of  Bolivia, 
and  the  sol  of  Peru  have  the  same  value. 

(c)  The  crowns  of  Norway,  Sweden,  and  Denmark  have  the  same  value. 

(d)  The  gourde  of  Hayti  and  the  peso  of  the  Argentine  Republic  have  the  same  value. 

(e)  The  anna  contains  12  pies, 


Art.  567.]  EXAMPLES.  233 


EXAMPLES. 

567.  1.  Find  the  cost  of  a  bill  of  exchange  on  London  for 
£225  at  4.81}.  (56O) 

ANALYSIS.— If  £1  costs  $4.81f,  £225  will  cost  225  times  $4.81£. 

2.  What  is  the  value  of  a  draft  for  £324  16*.  at  4.87J  ? 

ANALYSIS. — Write  one-half  of  the  greatest  even  number  of  shillings  as 
tenths  of  a  pound,  and  if  there  be  an  odd  shilling  write  5  hundredths. 
£324  16s.  =  £324.8.  (See  Art.  342,  Ex.  12,  Note.)  The  value  of  £324 16s. 
at  4.87£  is  found  by  multiplying  $4.87£  by  324.8. 

3,  Find  the  value  of  a  draft  on  London  for  £379  12*.  Id.,  at 
4.86|. 

OPERATION. 

379.6 

4.861 

ANALYSIS. — If  each  penny  be  regarded  as  2  cents,  the 

result  will  be  sufficiently  accurate.  For  lid.  the  maximum 
number  of  pence  in  any  example,  and  exchange  at  4.91,  the 
error  would  be  only  $  cent.  $4.86f  x*  379.6  =  $1846.28. 
$1846.28  +  $0.14  =  $1846.42.  To  save  one  addition,  add 
the  14  cents  to  the  partial  products  as  in  the  operation. 


1846.420 

Find  the  value  of  Find  the  value  of 

4.  £500  at  4.81£.  8.  £512  13*.  at  4.84|. 

5.  £775  at  4.85J.  9.  £834  6s.  Gd.  at  4.1 

6.  £837  at  4.83-J.  10.  £675  11*.  Sd.  at  4.87|. 

7.  £84  8s.  at  4.85.  11.   £225  7*.  5d.  at  4.82}. 

12.  Find  the  cost  of  a  bill  of  exchange  on  Liverpool,  for  £875 
12s.  Qd.  at  the  par  value.  (56O) 

18.  What  are  the  proceeds  of  a  draft  of  £959  5*.  4d.,  sold 
through  a  broker,  at  4. 79  J-,  brokerage  \%  ? 

14.  An  exporter  sold  a  draft  for  £540  3s.  on  Manchester,  pay- 
able in  London,  at  4. 84,  brokerage  \%.     What  were  the  proceeds  ? 

15.  Find  the  proceeds  of  a  draft  on  Newcastle-on-Tyne,  at  60 
days'  sight  for  £1764  15*.,  payable  in  London,  at  4.82,  brokerage 
on  exchange  \%. 

16.  An  importer  purchased  a  bill  of  exchange  on  London,  at  3 
days'  sight,  for  £488  16*.  6d.,  at  4.85J.     What  was  the  cost  ? 


234  EXCHANGE.  [Arl.  567. 

17.  How  much  exchange  on  London  at  4.81f  will  $821.99  buy  ? 

ANALYSIS. — $4.Slf  will  buy  exchange  for  £1  ;  hence,  $821.99  will  buy  as 
many  pounds  as  $4.81f  are  contained  in  $821.99,  or  £170.625.  £170.625  = 
£170  12s.  Qd.  (See  Art.  289,  and  Art.  342,  Ex.  19,  Note.) 

18.  What  will  be  the  face  of  a  3   days'  bill  of  exchange  on 
London  that  can  be  bought  for  $5964.13,  exchange  4.86|-  ? 

19.  The  face  of  a  bill  of  exchange  was  £875,  and  its  cost  was 
$4233.91.     What  was  the  rate  of  exchange  ? 

20.  An  exporter  received  $9063.22  for  a  bill  of  exchange  that 
was  sold  through  a  broker  at  $4.86J;  what  was  the  face  of  the 
bill,  the  broker's  commission  being  \%  ? 

21.  Find  the  cost  of  a  bill  of  exchange  on  Paris  for  7000  francs 
at  5.21J. 

OPERATION. 

5.21-J  )    7000  ANALYSIS. — Since  5.21|  francs   cost    $1, 

g  g  7000  francs  will  cost  as  many  dollars  as  5.21 1 

francs  are  contained  times  in  7000  francs. 
41.75    )  56000.0000  ( 

Find  the  value  of  Find  the  value  of 

22.  6000  francs  at  5.16.  25.  8475  francs  at  5. 19}. 
28.  5000  francs  at  5.18|.  26.   7216  francs  at  5.17}. 
24.  4000  francs  at  5.21|.  27.   987.60  francs  at  5.20|. 

28.  Find  the  cost  of  a  draft  on  Antwerp  at  3  days'  sight,  for 
9640  francs,  at  5.19f. 

29.  What  is  the  value  of  a  draft  on  London  for  £416  165.  3d., 
at  4.85|? 

SO.  Sold  exchange  on  Geneva,  through  a  broker,  for  8000 
francs  at  60  days'  sight ;  what  were  the  proceeds  of  the  draft, 
exchange  being  5.20f,  brokerage  \%  ? 

31.  What  are  the  proceeds  of  a  draft  on  Paris  for  12420  francs, 
at  5.19|,  brokerage  on  exchange  \%  ? 

82.  What  will  it  cost  to  remit  to  Antwerp  8750  francs  at  the 
par  value  ?     (561) 

83.  Sold  through  a  broker  a  draft  on  Geneva  for  7324  francs. 
What  were  the  proceeds,  exchange  being  5.18|,  brokerage  \%  ? 

84.  What  will  be  the  face  of  a  bill  of  exchange  on  Geneva  that 
can  be  bought  for  $15372,  exchange  selling  at  5.22J  ? 

85.  Paid  for  a  draft  on  Paris  $3460.32  ;  what  was  the  face  of 
the  draft,  exchange  being  5.19-f  ? 


Art.  567.]  FOREIGN     EXCHANGE.  235 

36.  A  merchant  paid  $6272  for  a  bill  of  exchange  of  32512.48 
francs  ;  what  was  the  rate  of  exchange  ? 

37.  Find  the   cost   of  a  bill   of  exchange  on  Hamburg  for 
14400  marks  (Reichsinarks)  at  94|. 

OPERATION. 
4  )   14400 

ANALYSIS.—  Since  4  marks  cost  $0.94|,  14400  marks  will 
cost  3600  (14400  -i-  4)  times  $0.94i,  or  $3388.50. 


3388.50 

Find  the  value  of 

38.  7200  marks  at  94.  41.  1237  marks  at  93|. 

39.  8416  marks  at  93£.  J$.  9894  marks  at  95|. 

40.  3456  marks  at  95J.  43.  6515  marks  at  94J  . 

44-  What  is  the  cost  of  a  bill  of  exchange  on  Frankfort  for 
16200  marks  at  95  £? 

45.  Sold  a  bill  of  exchange  on  Hamburg  for  13200  marks,  at 
94£  ;  what  was  the  amount  received,  brokerage  \%  ? 

46.  An  importer  purchased  a  bill  of  exchange  on  London  for 
£318  10s.  1d.9  at  4.85J  ;  what  did  it  cost  ? 

47.  What  were  the  proceeds  of  a  draft,  sold  through  a  broker,  f 
for  8748  marks,  at  94£,  brokerage  \%  ? 

48.  An  exporter  sold  a  draft  on  Paris  for  12275  francs,  at 
5.19|  ;  what  were  the  proceeds,  brokerage  \%  ? 

49.  What  is  the  face  of  a  bill  on  Hamburg  that  cost  $816, 
exchange  94J  ? 

ANALYSIS.  —  Since  $.94$-  will  buy  4  marks,  $816  will  buy  4  times  as  many 
marks  as  $0.94£  is  contained  times  in  $816. 

50.  What  is  the  face  of  a  3  days'  draft  on  Bremen,  that  was 
purchased  in  New  York  for  $3261.60,  exchange  94|  ? 

51.  The  cost  of  a  draft  of  12320  marks  was  $2922.15  ;  what 
was  the  rate  of  exchange  ? 

52.  Find  the  cost  of  a  bill  of  exchange  on  Amsterdam,  for 
7240  guilders,  at  40£. 

53.  Find  the  cost  of  a  bill  of  exchange  on  Amsterdam,  at 
GO  days'  sight,  for  12480  guilders,  exchange  39|. 

54.  An  exporter  received  $1890.86  for  a  bill  of  exchange  on 
Amsterdam;  what  was  its  face,  exchange  being  41£,  brokerage 


236  EXCHANGE.  [Art.  567. 

55.  At  40-f,  how  much  exchange  on  Amsterdam  will  $2877.93 
buy? 

56.  The  value  of  a  draft  of  5280  guilders  is  $2145  ;  what  is 
the  quotation  ? 

57.  The  dividends  of  the  N.  Y.  C.  and  H.  R  K.  Co.,  are  paid 
in  London  at  the  rate  of  49£  pence  to  the  dollar.     What  is  the 
equivalent  rate  of  exchange  ? 

58.  Find  the  value  in  U.  S.  money  of  16319  bushels  of  wheat 
at  45.  tyd.  per  bushel,  exchange  4.86J. 

59.  A  merchant  sent  a  messenger  with  a  bill  of  exchange  of 
20000  francs  to  two  bankers,  A  and  B,  with  instructions  to  sell  it 
to  the  best  advantage.     A  offered  5.27  and  B  5.27^.    The  messen- 
ger imprudently  accepted  the  latter  offer.     How  much  did  the 
merchant  lose  by  the  ignorance  of  the  messenger  ? 

60.  When  United  States  4  per  cent,  consols  are  quoted  in  New 
York  at  114J,  and  sterling  exchange  at  4.83J,  what  should  be  the 
London  quotation  of  the  bonds  ?   What  should  be  the  London  quo- 
tation of  4J  per  cent,  bonds,  the  New  York  quotation  being  113J  ? 

NOTE. — In  London,  all  American  securities  are  quoted  on  an  assumed 
value  of  the  pound  sterling  of  $5,  instead  of  the  actual  value  of  $4.8665,  or, 
more  definitely  speaking,  its  commercial  value  determined  by  the  rate  of 
exchange.  Multiplying  the  New  York  quotation  by  5  and  dividing  by  the 
rate  of  exchange,  the  result  will  be  the  equivalent  London  quotation. 

61.  When  American  railway  stocks  are  quoted  in  London  at 
88,  what  is  the  equivalent  New  York  quotation,  sterling  exchange 
being  quoted  in  New  York  at  4.88J  ? 

62.  What  is  the  London  equivalent  of  a  New  York  quotation 
of  142,  exchange  being  4.83  ? 

63.  At  Paris,  what  is  the  value  of  a  draft  on  London  of  £550, 
exchange  being  25.36J? 

64.  At  London,  what  is  the  cost  of  a  draft  on  Hamburg  of 
8000  marks,  exchange  being  20.45  ? 

65.  At  Vienna,  what  is  the  cost  of  a  draft  on  London  of  £625, 
exchange  being  11.75  ? 

66.  At  London,  what  is  the  value  of  a  draft  on  Calcutta  of 
12000  rupees,  exchange  being  quoted  at  Is.  S^d.  ? 

67.  A  commission  merchant  wishes  to  remit   $2475  to  his 
principal  in  England.     How  large  a  draft  must  he  purchase, 
exchange  being  4.83  J  ? 


EQUATION     OF    ACCOUNTS. 


568.  Equation  of  Accounts  (called  also  Equation  of  Pay- 
ments and  Averaging  Accounts)  is  the  process  of  finding  the  time 
when  several  debts  due  at  different  dates  may  be  paid  in  one 
amount  without  loss  of  interest  to  either  party.     It  is  also  the 
process  of  finding  the  time  when  the  balance  of  an  account  having 
both  debits  and  credits  may  be  paid  without  loss  of  interest  to 
either  party.     This  time  is  called  the  equated  or  average  time. 

NOTE.— It  is  important  that  the  commercial  student  be  thoroughly  drilled 
in  the  theory  and  practice  of  Equation  of  Accounts,  as  examples  in  this  sub- 
ject are  of  frequent  occurrence  in  many  wholesale  and  commission  houses. 

569.  To  find  the  equated  time  when  the  items  of  the 
account  are  all  on  the  same  side,  i.  e.,  all  debits  or  all 
credits. 

ANALYTICAL  STEPS. — By  assuming  a  certain  date  as  the  time  of  settle- 
ment, we  find  what  the  loss  or  gain  of  interest  would  be  to  the  payer  if  all 
the  bills  were  paid  by  him  on  that  date.  We  next  find  in  how  many  days  the 
total  amount  of  the  bills  would  produce  a  sum  equivalent  to  this  loss  or  gain  of 
interest,  and  find  the  true  day  of  settlement  by  counting  forward  or  back- 
ward this  number  of  days  from  the  assumed  date.  Thus,  if  the  sum  of  the 
several  bills  is  $1000,  and  the  loss  of  interest  to  the  payer  at  the  assumed 
date  of  settlement  is  $10  (the  interest  of  $1000  at  60  days  at  6%),  it  is  evident 
that  the  true  date  of  settlement,  or  the  time  when  there  would  be  no  loss  of 
interest  to  either  party,  must  be  60  days  after  the  assumed  date. 

NOTES.— 1.  The  interest  on  the  bills  paid  after  they  became  due  would 
equal  the  interest  on  the  bills  paid  in  advance,  the  former  being  a  gain  to  the 
payer,  and  the  latter,  a  loss. 

2.  Any  date  may  be  assumed  as  the  time  of  settlement.  For  convenience, 
the  earliest  or  latest  date  is  generally  used.  If  the  earliest  date  is  taken,  the 
estimated  interest  is  a  loss  to  the  payer ;  if  the  latest  is  taken,  the  interest  is 
again. 


238  EQUATION     OF     ACCOUNTS.  [Art,  569. 

When  the  time  is  found  by  Compound  Subtraction,  or  each  mouth  is 
regarded  as  30  days,  the  last  day  of  the  month  preceding  the  earliest  item  is 
the  most  convenient.  (See  second  interest  method.) 

In  Equation  Tables,  Dec.  31  or  Jan.  1  is  taken  for  all  examples. 

The  assumed  date  is  sometimes  called  the  focal  date. 

3.  Any  rate  of  interest  may  be  used  in  making  the  computations,  6  and 
12  being  the  most  convenient  rates. 

57O.  Ex.  At  what  date  may  the  following  bills  of  merchan- 
dise be  paid  in  one  amount  without  loss  of  interest  to  either 
party?  Due  Apr.  10,  $114;  due  Apr.  26,  $140;  due  May  22, 
$320 ;  due  June  6,  $976. 

OPEBATION. — PRODUCT  METHOD. 

Due  Apr.  10,     $114  x     0  =          0 

"     26,       140  x  16  =    2240 

"    May  22,       320  x  42  =  13440 

"    June  6,       976  x  57  =  55632 

1550  )  71312  (  46  days 

after  Apr.  10,  or  May  26. 

ANALYSIS.— For  convenience,  assume  Apr.  10,  the  earliest  due  date,  as 
the  time  of  settlement.  If  the  first  bill,  which  is  due  Apr.  10,  is  paid  on  that 
date,  there  will  be  no  loss  or  gain  of  interest  to  either  party.  If  the  second 
bill,  which  is  due  Apr.  26,  is  paid  Apr.  10,  16  days  before  it  is  due,  there  will 
be  a  loss  to  the  payer  of  the  interest  or  the  use  of  $140  for  16  days,  or  $2240  for 
1  day.  On  the  third  bill,  there  will  be  a  loss  of  the  interest  of  $320  for 
42  days,  or  $13440  for  1  day.  On  the  fourth  bill,  there  will  be  a  loss  of  the 
interest  of  $976  for  57  days,  or  $55632  for  1  day.  If  all  the  bills  are  paid 
Apr.  10,  there  will  be  a  loss  to  the  payer  of  the  interest  of  $71312  for  1  day, 
or  of  $1550  for  46  days.  Since  the  loss  of  interest  to  the  payer  is  equivalent 
to  the  interest  of  the  total  amount  of  the  bills  for  46  days,  it  is  evident  that 
the  day  when  there  would  be  no  loss  of  interest  must  be  46  days  after 
Apr.  10,  or  May  26.  The  payer  is  entitled  to  defer  payment  46  days  after  the 
assumed  date  as  a  compensation  for  the  estimated  loss. 

The  gain  of  interest  to  the  payer  on  the  first  three  bills,  which  are  paid 
after  they  are  due,  equals  the  loss  of  interest  on  the  fourth  bill,  which  is  paid 

before  it  is  due. 

PROOF. 

The  interest  of  $114  for  46  days  at  6^  is $0.874 

«          "      "    140  "   30     "          " 70 

«      "    320  "     4     "          "       .213 

Total  gain  of  interest  to  the  payer       1.787 

The  interest  (a  loss  to  the  payer)  of  $976  for  11  days  is  .       1.789 


Art.  570.]  EQUATION     OF    ACCOUNTS.  239 

NOTES. — 1.  In  finding  the  number  of  days  from  the  assumed  date  to  the 
other  dates,  instead  of  calculating  from  the  assumed  date  each  time,  find  the 
interval  from  one  date  to  the  next  and  add  it  to  the  last  number  of  days. 
Thus,  from  Apr.  10  to  May  22  is  42  days,  and  from  May  22  to  June  6,  15  days  ; 
hence,  from  Apr.  10  to  June  6  is  57  (42  +  15)  days.  (See  Art.  31O,  Ex.  3.) 

2.  To  determine  the  due  date,  find  the  number  of  days  in  the  operation 
nearest  to  the  quotient,  and  add  or  subtract,  as  may  be  necessary,  the  differ- 
ence between  it  and  the  quotient,  to  its  corresponding  date.  Thus,  in  the 
above  example,  the  number  of  days  in  the  operation  nearest  to  the  quotient  is 
42  ;  hence  the  due  date  is  4  (46-42)  days  after  May  22,  or  May  26.  (See  Art. 
311,  Ex.  10.) 

S.  If  the  fraction  of  the  quotient  is  less  than  -J-,  disregard  it  ;  if  more  than 
^,  add  1  day  to  the  integral  number  of  days  in  the  quotient. 

571.  RULE    FOR    THE    PRODUCT    METHOD. — Assume    the 
earliest  due  date  as  the  day  of  settlement  for  all  the  items. 
Multiply   each   item  by  the   number  of  days  intervening 
between  the  assumed  date  of  settlement  and  the  date  of  the 
item;  and  divide  the  sum  of  the  several  products  by  the 
sum  of  the  account.     Count  fonvard  from  the  assumed  date 
the  number  of  days  obtained  in  the  quotient.     TJie  result 
will  be  the  equated  time. 

572.  OPERATION. — FIRST  INTEREST  METHOD. 

Days.  Interest 


Due 

Apr. 

10, 

$114 

0 

$.00 

tt 

« 

26, 

140 

16 

(  - 

233 

for  10 

days. 

(  . 

14 

"   6 

(i 

« 

May 

22, 

320 

42 

(  1- 

60 
64 

"  30 
"  12 

a 

(4 

88 

"  30 

" 

tt 

Juno 

G, 

976 

57 

2, 

44 

"  15 

" 

60) 

15.50 

(l. 

952 

"  12 

(i 

.258         )  11.885  (  46  days 

after  Apr.  10,  or  May  26. 

ANALYSIS. — Assume  Apr.  10,  the  earliest  due  date,  as  the  time  of  settle- 
ment. If  the  total  amount  ($1550)  of  the  bills  is  paid  Apr.  10,  the  assumed  date 
of  settlement,  there  will  be  a  loss  of  interest  to  the  payer  of  $11.885.  The 
interest  of  $1550  for  60  days  at  6^  is  $15.50,  and  for  1  day,  $0.258.  It  will 
take  $1550  to  produce  $11.885  interest  as  many  days  as  $0.258  is  contained 
times  in  $11.885,  or  46  days.  If,  at  the  assumed  date  of  settlement,  there  is  a 
loss  to  the  payer  of  the  interest  of  $1550  for  46  days,  the  true  day  of  settle- 
ment must  be  46  days  later,  or  May  26. 


240  EQUATION     OF    ACCOUNTS.  [Art.  573 

573.  OPERATION. — SECOND  INTEREST  METHOD, 

Mo.  Days.  Interest. 

0  Apr.  10,     $114       $0.19 

j      .466  for  20  days. 

/CO.  J.4:U       < 

f    1.60      "      I  mo. 

1  May  22,      320    \    1.067    "    20  days. 

.107    "      2     " 


7.75    )  14.306  ( 1  wo.  25  da.  after  Mar.  31, 
7.75  or  May  25. 

6.556 

30 

7.75)  196. 680  (25  days. 
1550 


293 

ANALYSIS. — By  this  method,  the  last  day  of  the  month  preceding  the 
earliest  due  date  is  assumed  as  the  date  of  settlement,  and  the  time  is  found 
by  Compound  Subtraction,  each  month,  being  regarded  as  30  days. 

The  months  are  placed  on  the  margin  and  the  days  correspond  with  the 
number  of  days  in  the  given  dates. 

Mar.  31,  the  assumed  day  of  settlement,  there  is  a  loss  to  the  payer  of 
$14.306  interest,  or  the  interest  of  $1550  for  1  mo.  25  da.  The  equated  time 
is  therefore  1  mo.  25  da.  after  Mar.  31,  or  May  25. 

Since  this  method  regards  all  months  as  30  days  each,  its  results  are  not 
strictly  accurate.  The  error  in  this  example  is  1  day.  (See  preceding  results.) 

When  this  method  is  used,  and  accurate  results  are  required,  the 
necessary  corrections  may  be  made  by  adding  to  the  intervals  of  time  1  day 
for  each  intervening  month  containing  31  days.  If  the  month  of  February  is 
included,  2  days  should  be  subtracted  in  a  common  year  and  1  day  in  a  leap 
year. 

In  counting  forward  to  find  the  equated  time,  the  opposite  correction 
should  be  made.  Thus,  if  the  assumed  date  is  June  30  and  the  quotient  is 
2  mo.  20  da.,  the  equated  time  would  be  Sept.  18,  2  days  being  subtracted  for 
July  and  August. 

The  following  is  the  corrected  operation  for  the  given  example,  1  day 
being  added  to  the  time  of  the  fourth  item  for  the  month  of  May.  The  result 
is  the  same  as  by  the  product  and  the  first  interest  methods. 


Ait.  573.J  EQUATION     OF    ACCOUNTS.  241 

OPERATION. 

Mo.  Days.  Interest. 

0  Apr.  10,         $114       $0.19 

26  140  .466  for  20  days. 

'  "      6    " 

"      I  mo. 

1  May    22,          320    •     1.067    "    20  days. 

"      2    " 
"      2  mo. 

2  June     6  +  l,_976    <      .976    "      6  days. 

2 )  15.50    (      .162    "      1    " 

7.75    )  14. 468(1  mo.  26  r/0.  after  Mar.  31, 
7.75  or  May  26. 

6.718 

_30 

7.75)201.540(26  days. 

EXAMPLES. 

574.  1.  At  what  date  may  the  following  bills  be  paid  in  one 
amount  without  loss  of  interest  to  either  party  ?  Due  Sept.  10, 
$145;  Sept.  28,  $144;  Oct.  8,  $75 ;  Oct.  23,  $512. 

2.  What  is  the  equated  time  for  the  payment  of  the  following 
bills?    Due  Mar.  28,  $446;  May  3,   $212;  May  15,  $116;  May 
31,  $475  ;  June  12,  $345. 

3.  What  is  the  average  due  date  of  the  following  bills,  each 
being  due  at  the  date  given  ?    Jan.  5,  $127.85  ;  Jan.  26,  $134.18  ; 
Feb.  5,  $249.40  ;  Feb.  23,  $418.73  ;  Feb.  28,  $176.25. 

NOTE.— The  result  will  be  practically  the  same  if  the  nearest  dollar  is 
used  in  multiplying  or  in  calculating  the  interest.  Thus,  in  the  above 
example,  regard  the  amounts  as  128,  134,  249,  419,  and  176  respectively. 

When  there  are  several  items  in  the  example,  some  accountants  omit  the 
cents  and  units  of  dollars,  and  use  the  nearest  number  of  tens.  Thus,  if  the 
above  account  were  of  sufficient  length,  the  numbers  might  be  regarded  as 
13,  13,  25,  42,  and  18  respectively.  In  this  example  the  result  is  the  same, 
but  in  some  examples,  containing  the  same  number  of  items,  there  would  be 
a  discrepancy  of  one  or  more  days. 

4.  Sold  a  customer  bills  at  the  due  dates  and  to  the  amounts 
specified  :  June  1,  $152.73  ;  June  15,  $114.28 ;  July  16,  $247.84 ; 
July  25,  $88.90  ;  Aug.  18,  $735.42  ;  Aug.  29,  $416.34.     When  may 
the  whole  indebtedness  be  equitably  discharged  at  one  payment  ? 


242 


OF    ACCOUNTS. 


[Art.  574. 


5.  Average  the  following  account : 

NEW  YORK,  July  1,  1882. 
MESSRS.  KICE,  STIX  &  Co., 

To  LORD  &  TAYLOR,  Dr. 


1882. 

Apr.     4 

Mdse.  30  days  per  bill  rendered. 

1816 

37 

"      21 

"     30     "         "            "              .     . 

724 

25 

May  13 

(f         3Q        f(               «                     <C                         ^        ^ 

342 

46 

"      25 

"     30     "         "            "              .     . 

535 

84 

June  16 

"     30     "         "            "              '.     . 

628 

62 

Due  by  equation  June  *,  1882. 

***# 

** 

NOTE. — When  several  bills  are  sold  on  a  common  term  of  credit,  first 
find  the  average  date  of  purchase,  and  to  the  result  add  the  common  term  of 
credit. 

Certain  merchants  sell  uniformly  on  the  same  term  of  credit,  while  others 
sell  on  different  credits,  depending  upon  the  class  of  goods,  the  standing  of 
the  customer,  the  state  of  the  market,  etc.  (See  Art.  416.) 

6.  A.  Hamilton  bought  of  F.  A.  Leggett  &  Co.,  several  bills  of 
goods,  as  follows  : 

May  16,  a  bill  of  $212.46  on  60  days'  credit. 

"     28,        "         318.40  ((  60     " 
June    6,        "         275.64  "  60     "         « 
"   21,       •"         187.83   "  60     "         " 
July  13,       '"         835.60   "  60     "         " 

A  60-day  note  for  the  whole  amount  is  given  in  settlement. 
What  must  be  its  date,  no  allowance  being  made  for  the  days  of 
grace  ? 

7.  Sold  on  a  credit  of  90  days  the  following  bills  of  goods : 
Mar.   4,  $194.13;   Mar.   27,  $222.36;  Apr.  12,  $538.72;  May  3, 
$432.64;  May  28,  $303.10.     What  is  the  equated  time  of  pay- 
ment ?    How  much  will  settle  the  account  Aug.  1,  at  6%  ?    How 
much  July  1  ? 

NOTE. — When  monthly  statements  are  sent  to  customers  the  accounts 
are  frequently  averaged.  (See  Ex.  5.)  When  the  account  is  averaged,  the 
simplest  method  of  finding  the  cash  balance  due  at  a  certain  date,  is  to  cal- 
culate the  interest  on  the  total  amount  from  the  average  date  to  the  time  of 
payment,  and  add  it,  if  the  time  of  settlement  is  after  the  average  date,  and 
subtract  it,  if  before. 


Art.   574.]  EQUATION     OF     ACCOUNTS.  243 

Since  a  fraction  of  a  day  is  not  considered  in  determining  the  average 
date,  this  method  of  finding  the  cash  balance  is  not  as  accurate  as  that  of 
Art.  589,  in  which  the  interest  is  reckoned  on  each  item  separately. 

8.  A  commission  merchant  sold  several  bills  of  goods,  on  a 
credit  of  4  months,  as  follows  :    Aug.  16,  1881,  $387 ;  Sept.  4, 
1881,  $243.60;  Sept.  18,  1881,  $637.75;  Oct.  28,  1881,  $165.50; 
Dec.  10,  1881,  $856.45.     What  is  the  equated  time  of  payment  ? 

NOTE. — The  above  account  may  be  averaged  by  first  finding  the  average 
date  of  purchase,  and  adding  the  common  term  of  credit ;  or  by  finding  the 
due  date  of  each  bill  separately,  and  determining  the  average  due  date  from 
the  dates  thus  found.  Since  the  months  have  not  uniformly  the  same  num- 
ber of  days,  the  results  by  the  two  methods  sometimes  differ  by  one  or  .more 
days,  when  the  common  term  of  credit  is  expressed  in  months. 

9.  Bought  goods  on  6  months'  credit  as  follows  :    Feb.  16, 
1881,  $376.50;  Mar.  12,  1881,  $287.40;  Mar.  19,  1881,  $612.87; 
Apr.  5,  1881,  $345.60;   Apr.  26,  1881,  $134.80;   June  1,  1881, 
$612.35.     What  is  the  average  time  of  maturity  ?     How   much 
would  balance  the  account  Jan.  1, 1882  ?     How  much  Oct.  1, 1881  ? 

10.  Park  and  Tilford  sold  to  R.  M.  Bishop  &  Co.  the  following 
bills  of  merchandise  on  60  days'  credit:     Feb.  24,  $176.82;  Feb. 
28,  $327.49 ;  Mar.  16,  $282.75  ;  Mar.  28,  $512.14 ;  Apr.  7,  $438.36 ; 
Apr.  14,  $109.70  ;  May  1,  $632.65.     What  is  the  equated  time  of 
payment,  and  how  much  would  be  required  to  balance  the  account 
June  1  ?     How  much  July  1  ? 

11.  The  following  bills  of  merchandise  were  purchased  on  4 
months'  credit:    June  1,  $237.16;  June  18,  $146.75;  June  30, 
$333.84;  July  5,  $416;  July  16,  $535.62  ;  July  27,  $912.33  ;  Aug. 
13,  $345.60.     A  note  payable  in  4  months  was  given  in  settlement. 
What  was  its  date,  no  allowance   being  made  for  the  days  of 
grace? 

12.  Bought  goods  on  60  days'  credit  as  follows:     Aug.  11, 
$487.60  ;  Aug.  20,  $398.30  ;   Sept.  1,  $411.26  ;    Sept.  13,  $283.36  ; 
Sept.  22,  $112.43  ;   Sept.  30,  $555.55  ;    Oct.  20,  $342.48  ;   Nov.  4, 
$337.64.     What  is  the  average  due  date  ? 

13.  What  is  the  average  time  for  the  payment  of  the  following 
bills,  each  being  sold  on  a  credit  of  4  months  ?    Feb.  29,  $224.37; 
Mar.  13,  $642.50;   Mar.  31,  $377.65;    May  4,  $510.10;    May  19, 
$388.84;  June  3,  $476.25  ;  June  19,  $227.30  ;  Jure  30,  $562.75. 


244  EQUATION     OF    ACCOUNTS.  [Art.  574 

14-  Bought  several  bills  of  goods  as  stated  below  : 

June    3,  a  bill  of  $375  on  30  days'   credit. 

"     28,        "  420  "  60     "          " 

July  16,       "          560  "     4  months'" 
Sept     4,       "          228  "  90  days'       " 

What  is  the  equated  time  of  payment  ? 

NOTE. — When  the  bills  are  sold  on  different  terms  of  credit,  first  find  the 
due  date  of  each  bill  separately  as  in  the  following  operation. 

OPERATION. — PRODUCT  METHOD. 

Date  of  purchase.     Credit.  Due  date.       Amount.      Days.         Products. 

June  3,   30  days,  July  3,  $375  x   0  =     0 

"  28,   60  "   Aug.  27,   420  x  55  =  ***** 

July  16,   4  mo.,   Nov.  16,   560  x  ***  =  ***** 

Sept.  4,   90  days,  Dec.  3,   228  x  ***  —  ***** 

****        ******  ** 


OPERATION. — APPROXIMATE  INTEREST  METHOD.* 

Mo.  Days.  Credit.  Interest. 

0    June    3,     $375,     30  days,  ] 

(    4.20  "      2  mo. 

0  "      28,       420,     GO      "  1.68  "  24  days. 

(      .28  "      4     " 

11.20  "      4  mo. 

1  July   16,       £60,       4010.,      4    2'80  "      l    " 

.933  "  10  days. 

.56  "      6     " 


3     Sept.     4,       228,     90  days, 


r.915          7. 915  )  30.708  (  3  mo.  26  da.  after 
23.745     May  31,  or  Sept.  26, 
6.963 
30 

7. 915)  208.890  (26  days. 


*  See  second  interest  method,  Art.  573 


Art.  574.]  EQUATION     OF    ACCOUNTS.  245 

15.  What  is  the  equated  time  for  the  payment  of  the  following 
bills  ? 

July     5,  1882,  $516.60  on    4  months'  credit. 

28,     "       327.35    "   60  days' 
Aug.  15,     "        147.84    "     4  months'      " 
Sept.     8,     "        485.42    "  30  days'  " 

*       25,     "        230.39    "   60     "  « 

16.  Sold  several  bills  of  goods  as  follows : 

May     4,  a  bill  of  $418.75  on  30  days'  credit. 
"     16,       "  322.86    "    60     "  " 

June    1,       "  513.44    "      4  months'  " 

"     12,       «  118.70    «    60  days'       " 

"     30,       «  786.30    "      6  months'" 

July  16,       «  274.85    «    60  days'       « 

"What  is  the  average  time  of  payment,  and  how  much  would 
balance  the  account  Sept.  1  ?    How  much  Oct.  1  ? 

17.  What  is  the  average  time  of  maturity  for  the  payment  of 
the  following  bills  ? 

Mar.     4,  1883,  $117.26  on    4  months'  credit. 


it 

21,     " 

97.43 

a 

30 

days' 

a 

it 

29,     « 

243. 

84 

a 

60 

u 

u 

Apr. 

16,     " 

376. 

14 

n 

4 

months' 

a 

" 

30,     " 

182. 

75 

a 

90 

days' 

te 

May 

18,     « 

412. 

50 

ti 

60 

n 

(( 

June 

1,     " 

518. 

65 

n 

30 

te 

tf 

18.  Bought  goods  of  Henry  Welsh  as  follows  : 

Nov.  13,  1881,  a  bill  of  $138.42  on  30  days'   credit. 

"  30,     "  "  416.10  "  60     "           " 

Dec.  16,     "  "  324.70  "  30     " 

Jan.      5,  1882,  "  586.85  "  4  months' 

"  26,     "  «  234.38  "  60  days' 

Feb.  12,     "  "  93.60  "  4  months'  " 

"  23,     "  "  618.75  "  30  days' 

Mar.     5,     "  "  374.36  "  60     "           " 

What  is  the  equated  time  for  the  payment  of  the  whole? 


" 


246 


EQUATION     OF    ACCOUNTS. 


[Art.  574. 


19.  Average  the  following  sales  : 

Sept.     4,  1881,  $187.16  on     6  months'  credit 


(t 

16,      w 

332.40 

ft 

30 

days' 

« 

24,      « 

512.75 

(  ( 

6 

months' 

Oct. 

5,      - 

164.60 

a 

6 

a 

" 

27,      " 

187.30 

a 

6 

a 

Nov. 

5,      " 

436.75 

a 

60 

days' 

" 

16,      " 

126.00 

(t 

6 

months' 

20.  Average  the  following  account : 

Dec.  1,  1882,  $246.75  on  30  days7   credit. 

"  12,      "  312.40  "    60      " 

"  26,      "  819.46  "     4  months'    " 

Jan.  2,  1883,  674.32  "     4 

"  10,      "  126.60  "    60  days' 

Feb.  4,      "  434.50  "     4  months'    " 

575.  To  find  the  equated  time  for  the  payment  of  the 
balance  of  an  account  having  both  debit  and  credit  items. 

576.  Ex.    At  what  date  may  the  balance  of  the  following 
account  be  paid  without  loss  of  interest  to  either  party  ? 

Dr.      JOHN  ROACH  in  account  with  GEO.  H.  STUART.       Or. 


1882. 

1882. 

June    6 

Mdse.  30  da. 

456 

00 

July  26 

Cash. 

400 

00 

"     20 

"      60  da. 

384 

00 

Aug.  10 

t< 

375 

00 

July     5 

"        3  mo. 

216 

00 

."     10 

Mdse.  60  da. 

288 

00 

"     26 

"         3  mo. 

552 

00 

577.  OPERATION. — PRODUCT  METHOD. 

Due  Cr. 

July  26,  $400  x  20  = 


Due 

Dr. 

July 

6, 

1456 

X 

0 

=     0 

Aug. 

19, 

384 

x 

44 

=  16896 

Oct. 

5, 

216 

X 

91 

=  19656 

(6 

26, 

552 

x 

112 

=  61824 

1608 

98376 

1063 

48485 

8000 


545 


Aug.  10,     375  x  35  =   13125 

Oct.     9,     288  x  95  =  27360 

1063  48485 


)  49891  (  92  days  after  July  6,  or  Oct.  6. 


Art.  577.]  EQUATION     OF    ACCOUNTS.  247 

ANALYSIS. — First  find  the  due  date  of  each  item.  For  convenience,  as- 
sume July  6,  the  earliest  due  date,  as  the  day  of  settlement  for  all  the  items 
on  each  side  of  the  account.  (See  Art.  569,  Note  2.)  If  the  balance  of  the 
account  is  paid  July  6,  the  assumed  date  of  settlement,  there  would  be  a  loss 
to  the  payer,  on  the  debit  side  of  the  account,  equivalent  to  the  interest  of 
$98376  for  1  day,  and  a  gain  on  the  credit  side,  equivalent  to  the  interest  of 
$48485  for  1  day;  or  a  net  loss  of  $49891  for  1  day,  or  of  $545  for  92  days. 
Since  the  loss  of  interest  to  the  payer  by  settling  the  account  July  6,  is 
equivalent  to  the  interest  of  the  balance,  or  the  amount  paid,  for  92  days,  it  is 
evident  that  the  day  when  there  would  be  no  loss  of  interest  must  be  92  days 
after  July  6,  1882,  or  Oct.  6,  1882. 

If  the  greater  sum  of  tb.e  products  had  been  on  the  credit  side,  there 
would  have  been  a  gain  to  the  payer  by  settling  the  account  July  6,  and  the 
day  that  the  balance  of  the  account  would  commence  to  draw  interest  would 
have  been  92  days  before  July  6,  or  Apr.  5, 1882. 

578.  EULE  FOR  THE  PRODUCT  METHOD. — First  find  the 
due  date  of  each  item.  Assume  the  earliest  due  date  as  the 
day  of  settlement  for  all  the  items  on  both  sides  of  the  ac- 
count. Multiply  each  item  by  the  number  of  days  inter- 
vening between  the  assumed  date  of  settlement  and  the  due 
date  of  the  item,  and  find  the  sum  of  the  products  on  each 
side  of  the  account.  Divide  the  balance  (the  difference  be- 
tween the  sums  of  the  debit  and  credit  products)  of  the 
products  by  the  balance  of  the  account.  The  quotient  will 
be  the  number  of  days  intervening  between  the  assumed 
date  and  the  true  date  of  settlement. 

To  find  the  true  date  of  settlement,  count  forward  from 
the  assumed  date,  when  the  balance  of  the  account  and  the 
balance  of  the  products  are  on  the  same  side  (both  debits  or 
both  credits) ;  and  count  backward,  when  on  opposite  sides. 

NOTES. — 1.  The  rule  for  counting  backward  and  forward  is  the  reverse  of 
the  above,  when  the  latest  date  or  a  date  after  the  latest  date  is  taken  as  the 
assumed  date  of  settlement. 

2.  Although  the  principles  of  equation  of  accounts  are  theoretically  correct, 
they  are  not  always  practicable  and  can  not  be  legally  enforced.     Thus,  if  a 
debt  of  $4000  is  due  Feb.  1,  no  merchant  would  accept  a  payment  of  $3600, 
Jan.  \,  with  the  understanding    that  the  remaining   $400  would  remain 
unsettled  9  months  after  Feb.  1,   or  until  Nov.  1.     The  merchant  would 
undoubtedly  be  willing  to  allow  a  discount  equivalent  to  the  interest  of  $3600 
for  the  unexpired  time,  or  1  month. 

3.  In  finding  the  equated  time,  reject  the  cents  when  less  than  50 ;  and 
add  1  dollar  to  the  dollars  when  the  cents  are  more  than  50.     The  results  will 
be  sufficiently  accurate. 


248 


EQUATION     OF    ACCOUNTS. 


[Art.  579. 


579.  OPERATION. — FIRST  INTEREST  METHOD.* 


Dr. 


Cr. 


Due 

July     6, 

Aug.  19, 

Oct.      5, 

"      26, 


$456 
384 
216 
552 

Days. 
•   0 

44 
91 
112 

Interest. 
$0.00 

2.816 
3.276 
10.304 

1608 
1063 
)5.45 

16.396 
8.08 
)  8.316< 

Due 

July  26, 
Aug.  10, 
Oct.  9, 


375 

_2S8 

1063 


Days. 

20 
35 
95 


Interest. 

$1.333 

2.187 
456_ 

8.080 


8.3160  (  92  days  after  July  6,  or 
.0908  Oct.  6,  1882. 

ANALYSIS.— If  the  account  is  settled  July  6,  the  assumed  date  of  settle- 
ment, Mr.  R.  would  be  entitled  to  a  discount  on  the  debit  side  of  $16.396,  and 
Mr.  S.  on  the  credit  side  of  $8.08;  or,  Mr.  R.  would  be  entitled  to  a  net  dis- 
count of  $8.316.  If,  by  paying  the  balance  of  the  account,  July  6,  Mr.  R.  is 
entitled  to  a  discount  of  $8.316,  it  is  evident  that  he  should  be  allowed  to 
defer  payment  until  the  balance  would  produce  an  equivalent  interest,  or  92 
days.  Hence,  the  true  date  of  settlement  is  92  days  after  July  6,  1882,  or 
Oct.  6, 1882. 

When  the  balance  of  the  account  and  the  balance  of  interest  are  both  due 
the  same  party,  the  equated  time  is  previous  to  the  assumed  date  of  settle- 
ment ;  and,  when  the  balance  of  the  account  and  the  balance  of  interest  are 
due  different  parties,  the  equated  time  is  after  the  assumed  date. 

58O.  In  the  following  operation,  the  latest  due  date  is  assumed 
as  the  date  of  settlement  for  all  the  items : 


OPERATION. 


Due 

Days. 

Interest. 

July     6, 

$456 

112 

$8.512 

Aug.  19, 

384 

68 

4.352 

Oct.      5, 

216 

21 

.756 

"      26, 

552 

0 

.00 

1608 

13.620 

1063 

11.761 

Due  Days.  Interest 

July   26,  $400  92  $6.133 

Aug.   10,  375  77  4.812 

Oct.      9,  288  17  .816 


1063 


11.761 


60)5.45     .0908)1.8590  (  20  days  before  Oct.  26,  or 

.0908  Oct.  6,  1882. 

ANALYSIS. — If  the  account  is  settled  Oct.  26,  the  assumed  date  of  settle- 
ment, the  payer  will  be  obliged  to  pay  $1.859  interest  in  addition  to  the 
balance  of  the  account.  Hence,  the  date  when  the  balance  only  may  be  paid 
without  loss  to  either  party  must  be  20  days  before  Oct.  26, 1882,  or  Oct.  6, 1882. 


*  See  Art.  5  7  3. 


Art.  581.] 


EQUATION     OF    ACCOUNTS. 


249 


581.  OPERATION. — APPROXIMATE  INTEREST  METHOD.* 


Dr. 


Cr. 


Mo. 


Days. 


Credit.          Interest. 


0  June    6,  $456  30  da. 


|  $2.2$ 
t      .456 


{3 
i.: 

1  July     5,    216    3  mo.  j    4'^ 
(      .lo 


3.84 
,28 
.32 

18 

r  n.04 

26,    552    3  mo.  -j    1.84 
V      .552 

25.788 
14.348 


Mo. 


Days. 


Credit.          Interest 

r$2.oo 
1.333 
I     .40 
j-    3.75 
(      .625 

10,    288  60  da.  j 

1063  14.348 


1  July  26,  $400 


2  Aug.  10,    375 


1608 
1063 


2  )  5.45     2.725  )  11.440  (  4  mo.  6  da.  after  May  31,  or 
2.725  10.900  Oct.  6. 

.540 
30 


2.725  )  16.200  (  6  days. 


EXAMPLES. 


582.     1.  At   what   date   may  the  balance  of  the  following 
account  be  paid  without  loss  to  either  party  ? 


Dr. 


ISAIAH  B.  PRICE. 


Cr. 


1889. 

May  16 
"      31 

To  Mdse. 

(t       « 

437 
324 

00 
00 

1889. 

May  23 
June  16 

By  Cash. 

a      a 

400 
300 

00 
00 

2.  Find  the  average  date  of  maturity  for  the  balance  of  the 
following  account : 


Dr. 


WILLIAM  C.  DOUGLAS. 


Cr. 


1888. 

1888. 

Jan.    4 

Mdse.  30  da. 

516 

00 

Feb.   1 

Cash.    .    . 

500 

00 

"     28 

60  da. 

325 

00 

"      1 

Note  60  da. 

300 

00 

Feb.    4 

"         4  mo. 

437 

00 

(63  da.) 

*  See  second  interest  method,  Art.  573,  and  second  method,  Ex.  14,  page  244. 


250 


EQUATION     OF    ACCOUNTS. 


3.  Average  the  following  account : 
Dr.  JOSEPH  H.  WEIGHT. 


[Art.  582. 


Cr. 


1882. 

Mar.  27 

Mdse,    4  mo. 

716 

48 

1882. 

Apr.  16 

Cash.    .     . 

300 

Apr.  16 

"       60  da. 

325 

75 

May    2 

tt 

400 

May     1 

"         4  mo. 

413 

40  ' 

July    8 

K 

500 

June    4 

"         4  mo. 

716 

87 

4>  What  is  the  equated  time  for  the  payment  of  the  balance  of 
tho  following  account  ? 

Dr.  A  in  account  with  B.  Cr. 


1882. 

Mar.  16 

Mdse.    4  mo. 

444 

57 

1882, 

July     1 

Cash.      .     . 

400 

"      30 

"      60  da. 

376 

82 

"      20 

t( 

375 

Apr.  20 

"     30  da. 

712. 

19 

Aug.  16 

re 

700 

May  17 

"       4  mo. 

628 

75 

"     30 

a 

600 

"     28 

"       4  mo. 

419 

31 

5.  Average  the  following  account.     What  will  be  the  amount 
due  Jan.  1,  1882  ? 
Dr.  C  in  account  with  D.  Cr. 


1881. 

1881. 

Jun3  16 

Mdse.  30  da. 

517 

25 

June  16 

Note60(63)<7«. 

1000 

"    28 

"      60  da. 

487 

50 

July  30 

Cash.       .     . 

375 

July     5 

4  mo. 

816 

75 

Aug.  13 

Mdse.  4  mo. 

900 

"     21 

"        6  mo. 

924 

30 

Oct.     5 

Cash.       .     . 

500 

Aug.  12 

"       4  mo. 

317 

65 

6.  When  will  the  balance  of  the  following  account  commence 
drawing  interest  ?     How  much  would  be  due  Mar.  1,  1883. 

Dr.  ANDREW  CARNEGIE,  Pittsburg,  Pa.  Cr. 


1882. 

Sept.     4 

Cash 

100 

1882. 

Aug.  16 

Mdse.     4  mo. 

647 

13 

"        4 

Note  4  mo. 

900 

"      29 

"         4  mo. 

322 

85 

Oct.     31 

Cash 

250 

Sept.     4 

4  mo. 

412 

90 

Dec.    28 

(( 

600 

"      17 

4  mo. 

588 

33 

"      17 

30  da. 

246 

12 

Nov.     4 

4  mo. 

683 

45 

Art.  582.]    EQUATION     OF    ACCOUNTS     SALES. 


251 


7.  Find  the  equated  time  for  the  payment  of  the  balance  of  the 
following  account. . 

Dr.  JAMES  B.  FARWELL,  Chicago,  111.  Or. 


1881. 

1881. 

Jan.      4 

Mdse.    4  mo. 

637 

20 

Mar.   16 

Cash. 

300 

00 

«       14 

"         4  mo. 

412 

87 

Apr,    20 

u 

400 

00 

«       14 

60  da. 

214 

35 

May      3 

if 

200 

00 

Mar.    16 

"         4  mo. 

298 

60 

3 

Note  4  mo. 

800 

00 

"       28 

30  da. 

973 

25 

8.  Average  the  following  account : 
Dr.  ABNOLD,  CONSTABLE,  &  Co. 


Or. 


1882. 

• 

1882. 

Apr.      4 

Mdse.     4  mo. 

426 

32 

Apr.    25 

Cash. 

375 

"       20 

"       Cash. 

387 

40 

June  30 

u 

600 

May    13 

60  da. 

622 

39 

July    31 

Note  60  da. 

600 

"      27 

"       30  da. 

584 

75 

Aug.  15 

Cash. 

500 

July     5 

4  mo. 

224 

50 

Oct.    31 

tt 

400 

"       16 

"         4  mo. 

838 

95 

' 

583.  To  find  the  equated  time  for  the  payment  of  the 
net  proceeds  (423)  of  an  account  sales  (434). 

584.  1.  The  sales  form  the  credit  side  of  the  account,  and 
the  charges  and  advances  the  debit  side. 

2.  The  charges  for  transportation,  cartage,  and  other  items 
paid  by  the  commission  merchant  are  considered  due  at  the  time 
of  the  payment  of  the  same. 

3.  The  commission  and  other  after-charges  of  the  commission 
merchant  are  considered  due  by  some  at  the  average  due  date  of 
the  sales ;  and  by  others,  at  the  average  date  of  the  sales.     Since 
the  commission  is  so  small  compared  with  the  gross  sales,  in  many 
examples,  it  makes  no  difference  which  date  the  commission  is 
considered  due.     Certain  merchants  enter  the  commission  at  the 
date  the  account  sales  is  rendered,  and,  by  so  doing,  produce  a 
result  sufficiently  accurate. 

4.  Many  commission  merchants,  when  the  consignments  are  not 
separated  and  numbered,  enter  the  sales  and  commission  only  on 
the  account  sales  (See  Ex.  4,  Art.  586),  and  enter  the  advances 


252 


EQUATION     OF    ACCOUNTS. 


[Art.  584. 


and  the  general  charges  in  the  account  current  (See  Ex.  6,  Art. 
594).  Accounts  sales,  when  the  shipments  are  continuous,  are 
rendered  monthly  to  the  manufacturers  or  consignors,  and 
"sketches  "  weekly  or  whenever  a  sale  is  made. 

5.  With  the  exception  of  finding  the  date  for  the  commission 
and  other  after-charges,  the  process  of  averaging  an  account  sales 
is  exactly  the  same  as  that  of  averaging  an  account  containing 
both  debit  and  credit  items. 

585.  Ex.  What  is  the  equated  time  for  the  payment  of  the 
net  proceeds  of  the  following  account  sales  ? 

NEW  YORK,  Dec.  1,  1881. 
Account  sales  of  Seed 

For  account  of  WILLIAM  STEPHENS  &  Co. 
By  FRANKLIN  EDSO^  &  Co. 


1881. 

Nov. 

4 

45^-  lu.  Timothy  Seed    .     30  da. 

13JL 

79 

53 

tt 

18 

50       "    Mammoth  Cl.  Seed  60  da. 

9iiJL 

450 

t( 

28 

49JJ1   «    Clover  Seed    .     .     Cash. 

SASL 

418 

32 

947 

85 

CHARGES. 

Oct. 

31 

Transportation  

60 

00 

Dec. 

1 

Commission  5%  as  Dec.  22,  1881. 

. 

47 

39 

107 

39 

Net  proceeds  due  Dec.  26,  1881.    . 

. 

840 

40 

ANALYSTS. — The  average  due  date  of  the  sales  is  Dec.  22, 1881,  which  is 
taken  as  the  due  date  for  the  commission. 

The  account  sales  to  be  averaged  will  now  be  as  follows  : 


Dr. 

Due   Oct.    31,   1881, 
"     Dec.  22,      " 


Cr. 

$60.00  Due   Dec.     4,    1881,  $79,53 

47.39  "     Jan.    17,    1882,  450.00 

"     Nov.  28,   1881,  418.32 

By  averaging  the  above,  we  find  the  net  proceeds,  $840.46,  are  due  Dec. 
26,  1881. 

If  the  commission  is  considered  due  Nov.  21,  1881,  the  average  date  of 
the  sales,  the  net  proceeds  will  be  due  Dec.  28,  1881. 

NOTE. — If  the  same  assumed  date,  or  focal  date,  be  taken  in  finding  the 
average  due  date  of  the  sales  as  in  finding  the  average  due  date  of  the  net 
proceeds,  the  operation  of  the  former  will  form  the  credit  side  of  the  latter 
operation. 


Art.  586.]    EQUATION     OF    ACCOUNTS     SALES. 


253 


EXAMPLES. 

586.  Find  the  net  proceeds  and  equated  time  of  the  following 
accounts  sales.  (Unless  otherwise  stated,  the  commission  is  con- 
sidered due  at  the  average  due  date  of  the  sales.) 

1.  Sales  of  400  bbls.  flour  received  per  N.  Y.  C.  &  H.  R.  E.  K., 
for  account  of  A.  W.  ARCHIBALD,  Ottumwa,  Iowa. 


1881. 

May 

11 

125  bbls.  "  Kirkwood  "  cash,  .     .      6^. 

*** 

** 

({ 

12 

150     "      "Iowa"          4  mo.,      .      6^ 

*** 

« 

18 

125     "     "Kirkwood  "4  mo.,      .      7^ 

*** 

***# 

** 

CHARGES. 

May 

3 

Transportation  and  Cartage,      .     .    . 

425 

tt 

4 

15 

(( 

18 

Storage,  , 

45 

Commission  and  Guaranty  5$,  .     .     . 

##* 

#* 

**# 

** 

Net  proceeds  due  per  average,  ,  1881, 

*##* 

** 

E.  &  0.  E.                                                                          E.  R  LlVERMORE. 

NEW  YORK,  May  20,  1881. 

What  would  be  the  equated  time  for  the  payment  of  the 
above  proceeds,  if  the  commission  and  guaranty  were  considered 
due  at  the  average  due  date  of  the  sales  ?  At  the  average  date  of 
the  sales  ?  If  considered  due  May  18,  the  date  of  the  last  sale  ? 

2.  Account  sales  of  900  sides  hemlock  sole  leather  by  MAS- 
SET  &  JACKET,  for  account  of  GRANT  &  HORTON,  Ridgway,  Pa. 


1881. 

Aug. 

ft 

a 

14 

18 
21 

Sides. 

400" 
300 
200 

Description. 

Terms. 

Weight. 

Price. 

w 

27J 
27i 

| 
*#** 
**** 
**** 

** 
** 
** 

**** 

** 

"Ridgway"  #7 

87 

88 

4  mo. 
4  mo. 
30  da. 

9407 
6875 
4712 

CHARGES. 

Aug. 
tt 

2 
3 

Transportation  $70,  Cartage  $9,  .     . 
Inspection,      
Commission  and  Guaranty  5$,      .    . 
Proceeds  due  ,  1881,     .... 

** 

9 
*** 

** 

*#* 

** 
** 

**** 

E.  &  0.  E. 

MASSEY  &  JACKET. 

PHILADELPHIA,  PA.,  Aug.  22,  1881. 

EQUATION     OF    ACCOUNTS. 


[Art.  586. 


3.  Find  the  equated  time  for  the  payment  of  the  net  proceeds 
of  Ex.  25,  Art.  427,  supposing  that  the  merchandise  was  sold  for 
cash,  and  that  the  commission  was  due  at  the  date  given. 

4.  Sales  by  JAMES  TALCOTT,  New  York,  for  account  of  Phenix 
Mills,  Cohoes,  N.  Y.     March  31,  1882.* 


Date. 

Cases. 

No. 

Description. 

Time. 

Yards. 

Price. 

Amount. 

Mar.    1 

2 

7619 

Fancy  Cassimere. 

30  da. 

9662 

1.35 

****** 

"     10 

4 

3475 

<(            t( 

10  da. 

1994 

1.70 

****** 

"    13 

3 

4157 

te            t( 

30  da. 

15061 

2.30 

****** 

"     17 

4 

6283 

(t            (( 

4  mo. 

19363 

1.65 

****** 

"    26 

2 

3971 

(f            t( 

Cash. 

978 

1.85 

****  ** 

Less  Commission  5$, 
Proceeds  due ,  1882, 


*****  ** 
***  ** 


*****  ** 


5.  Account    Sales    of   merchandise  by  Jonif  F.  COOK,    for 
account  of  Excelsior  Packing  Co.,  Cincinnati,  Ohio. 


1881. 

Oct. 

16 

50  Bbls.  Mess  Beef,      .     .     Cash. 

HJLJL 

*** 

** 

it 

24 

100     "     N.  M.  Pork,     .     . 

17JJL 

**** 

" 

31 

25     "     Hams  6376  Ibs.,  .     10  da. 

13** 

*** 

** 

Nov. 

9 

25     "     Shoulders  5717  Ibs.,  60  da. 

90 

*** 

** 

tt 

18 

75     "     C.  M.  Pork,     .     .       4  mo. 

13H 

**** 

** 

**** 

** 

CHARGES. 

Oct. 

13 

Transportation,    

325 

15 

Cartasre, 

37 

50 

to 

15 

Cooperasre, 

15 

„ 

15 

Inspection,      

13 

75 

Nov 

18 

Storage, 

48 

Commission  5^,  

*** 

** 

*** 

** 

E.  &  0.  E. 

NEW  YORK,  N.  Y.,  Nov.  20,  1881. 


JOHN-  F.  COOK. 


*  If  the  commission  is  considered  due  at  the  average  due  date  of  the  sales,  and  since 
there  are  no  other  changes,  the  net  proceeds  will  he  due  at  the  same  date. 


ACCOUNTS     CURRENT. 


587.  An  Account  Current  is  an  itemized  account  of  the 
business  transactions  between  two  houses,  showing  the  balance  or 
amount  due  at  the  current  date.  The  amount  due  is  sometimes 
called  the  cash  balance. 

1.  An  account  current  is  a  transcript  of  the  ledger  account 
with  the  addition  of  certain  details  taken  from  the   books  of 
original  entry,  and  is  arranged  in  a  different  form. 

2.  Interest  is  charged,  or  not,  according  to  the  custom  of  the 
business,  or  the  agreement  between  the  parties.      This  chapter 
treats  only  of  accounts  in  which  interest  is  charged.     When  inter- 
est is  not  charged,  the  balance  due  is  the  difference  between  the 
two  sides  of  the  account  as  originally  entered  in  the  ledger.     The 
interest  may  be  reckoned  according  to  any  of  the  methods  of  Art. 
437.      In  the  illustrative  example   the  exact  time  in  days   is 
found,  and  the  days  are  regarded  as  360ths  of  a  year.    In  the 
examples   for  practice,  unless  otherwise   stated,   the   interest   is 
reckoned  on  the  same  basis. 

3.  Accounts  current  are  rendered  by  merchants,  bankers,  and 
brokers    annually    (Ex.    2),    semi-annually    (Ex.    1),     quarterly 
(Ex.  3),  or  monthly  (Ex.  6).      Since  the  interest  draws  interest 
after  the  account  is  balanced,  the  oftener  the  account  is  balanced, 
or  the  interest  is  added  to  the  account,  the  greater  the  amount 
due.     Some  merchants  render  partial  accounts  current  monthly, 
but  do  not  carry  the  interest  to  the  main  column  until  the  end  of 
the  year  (Ex.  11).    The  twelve  partial  accounts  current  make,  when 
combined,  the  complete  account  current  for  the  whole  year. 

4'  There  are  three  methods  in  common  use  for  finding  the 
amount  due  on  an  account,  including  interest,  at  a  certain  date, 
all  of  which  are  presented  in  the  following  illustrative  example  : 
1.  By  interest ;  2.  By  products  ;  3.  By  daily  balances. 


256 


ACCOUNTS     CURRENT. 


[Art.  588. 


588.  Ex.     Find  the  amount  due,  including  interest  at  6%,  on 
the  following  account  Jan.  1,  1882. 

Dr.        GEO.  W.  CHILDS  in  account  with  A.  A.  Low.        Cr. 


1881. 

1881. 

— 

Oct.     1 

Balance. 

1800 

Oct.  31 

Cash. 

1000 

"    16 

Mdse.  30  da. 

360 

Nov.  16 

NoteSOdo. 

600 

Nov.  27 

30  da. 

432 

Dec.    4 

Cash. 

240 

Dec.  18 

BillofH.C.&Co. 

420 

"     26 

tt 

300 

589.  OPERATION. — INTEREST  METHOD. 


Due. 


Oct.  1, 

Nov.  15, 

Dec.  27, 

"  18, 


Dr. 

Amount. 

$1800 
360 
432 
420 

$3012 
2140 


92 

47 

5 

14 


Interest. 

Due. 

$27.60 

Oct.    31, 

2.82 

Dec.   19, 

.36 

"       4, 

.98 

"     26, 

$31.76 

13.05 

Or. 

Amount.     Days.       Interest. 


$1000 
600 
240 
300 

$2140 


62 

13 

28 

6 


$10.33 

1.30 

1.12 

_^3p 

$13.05 


872   +   18.71  =  890.71. 


ANALYSIS. — First  find  the  due  date  of  each  item  of  the  account.  Each 
item  will  draw  interest  from  its  due  date  until  the  day  of  settlement,  or  Jan. 
1,  1882.  The  total  interest  on  the  debit  side  of  the  account  is  $31.76,  and  on 
the  credit  side,  $13.05.  The  balance  of  interest,  $18.71,  is  therefore  in  favor 
of  the  debit  side,  or  is  due  Mr.  Low. 

Since  both  the  balance  of  the  account  ($872)  and  the  balance  of  interest 
($18.71)  are  due  the  same  party,  the  net  amount  due  Jan.  1,  1882,  is  $872  plus 
$18.71,  or  $890.71. 

If  the  balance  of  interest  had  been  on  the  credit  side  of  the  account,  the 
net  amount  due  would  have  been  $872  minus  $18.71,  or  $853.29. 

NOTES. — 1.  It  will  sometimes  happen  that  certain  items  will  fall  due 
after  the  day  of  settlement.  The  interest  on  such  items  should  be  transferred 
to  the  opposite  side  of  the  account.  (See  Ex.  8.) 

2.  If  the  account  has  been  averaged,  the  amount  due  at  a  given  date  may 
be  found  by  calculating  the  interest  on  the  balance  of  the  account  from  the 
time  it  is  due  to  the  date  of  settlement.     If  the  date  of  settlement  is  earlier 
than  the  average  date,  subtract  the  interest  from  the  balance  of  the  account ; 
if  later  than  the  average  date,  add  the  interest.    (See  Art.  574,  Ex.  7,  Note.) 

3.  The  interest  method  is  generally  used  in  business.     Since  it  gives  the 
interest  on  each  item  and  is  readily  understood,  it  is  more  satisfactory  to  those 
to  whom  accounts  current  are  sent  than  the  product  method.    When  interest 
tables  are  used,  it  is  shorter  than  any  other  method. 


Art.  590.] 


ACCOUNTS     CURRENT. 


257 


59O.  The  following  is  a  common  form  of  an  account  current 
including  interest : 

Dr.       GEO.  W.  CHILDS  in  %  current  with  A.  A.  Low.       Cr. 


1881. 

bays. 

Interest. 

Amounts. 

1881. 

Days. 

Interest. 

Amounts. 

Oct.     1 

Balance. 

92 

27.60 

1800.00 

Oct.  31 

Cash. 

62 

10.33 

1000.00 

"     16 

Mdse.  as  Nov.  15. 

47 

2.82 

360.00 

Nov.  16 

Note  as  Dec.  19. 

13 

1.30 

600.00 

Xov.27 

"  Dec.  27. 

5 

.36 

432.00 

Dec.    4 

Cash. 

28 

1.12 

240.00 

Dec.  18 

BillofH.C.  &Co. 

14 

.98 

420.00 

"     26 

" 

6 

.30 

300.00 

1882. 

1882. 

Jan.    1 

Bal.  of  Interest. 

18.71 

Jan.    1 

Bal.  of  Interest. 

18.71 

1 

"     "  Account. 

890.71 

1830. 

31.76 

3030.71 

31.76 

3030.71 

Jan.    1 

Balance. 

890.71 

591.  KULE  FOR  THE  INTEREST  METHOD. — First  find  the 
due  date  of  each  item  of  the  account.  Then  find  the  inter- 
est on  each  item  from  the  date  it  becomes  due  to  the  day  of 
settlement.  The  difference  between  the  sums  of  the  debit 
and  the  credit  interest  will  be  the  balance  of  interest. 

To  find  the  net  amount  due,  when  the  balance  of  interest 
and  the  balance  of  items  are  on  the  same  side,  take  their 
sinn ;  ivhen  on  opposite  sides,  talce  their  difference. 


593.  OPERATION. — PRODUCT  METHOD. 


Due. 


Dr. 
Am't.        Days.        Products. 


Cr. 


Due. 


Am't. 


Oct. 

Nov.  15,       360  x  47  = 
27,       432  x    5  = 
420  x  14  = 


Days.       Products. 


Dec. 


IB, 


$3012 
2140 

~872 


165600 

Oct. 

31, 

$1000 

X 

62  = 

62000 

16920 

Dec. 

19, 

600 

X 

13  = 

7800 

2160 

a 

4, 

240 

X 

28  = 

6720 

5880 

tt 

26, 

300 

X 

6  = 

1800 

190560 

$2140 

78320 

78320 


$872  +  $18.71  =  $890.71. 


6  )  112240 

$18.706 

ANALYSIS. — By  multiplying  the  number  of  dollars  by  the  number  of  days, 
and  taking  the  sum  of  the  products  on  each  side  of  the  account,  we  find  that 
the  total  debit  interest  is  equivalent  to  the  interest  of  $190560  for  1  day,  and 
the  total  credit  interest  to  the  interest  of  $78320  for  1  day.  The  balance  of 
interest  is  therefore  equivalent  to  the  interest  of  $112240  for  1  day.  The 
interest  of  $1  for  1  day  is  £  of  a  mill  (446),  and  of  $112240,  18706  (i  of  112240) 
mills,  or  $18.71.  Since  the  balance  of  items  ($872)  and  the  balance  of  interest 
($18.71)  are  both  due  the  same  party,  the  net  amount  due  is  their  sum,  or  $890.71. 


258  ACCOUNTS      CURRENT. 

593.  OPERATION. — BY  DAILY  BALANCES. 


[Art.  593. 


Date. 

Dr. 

Cr. 

Dr.  Balances. 

Days. 

Dr.  Products. 

Oct.  1 

1800 

1800 

30 

54000 

"   31 

1000 

800 

15 

12000 

Nov.  15 

360 

1160 

19 

22040 

Dec.  4 

240 

920 

14 

12880 

"  18 

420 

1340 

1 

1340 

"   19 

600 

740 

7 

5180 

"   26 

300 

440 

1 

440 

"   27 

432 

872 

5 

4360 

3012 

2140 

92 

6  )  112240 

2140                           18.706 

872  +  18.71  =  890.71. 

ANALYSIS. — Arrange  the  debit  and  the  credit  items  in  the  order  of  their 
dates  as  in  the  operation.  Find  the  balance  of  the  items  at  each  of  the  dates. 
There  is  a  debit  balance  of  $1800  for  30  days  ;  the  interest  of  which  is  equiv- 
alent to  the  interest  of  $54000  for  1  day.  The  interest  of  the  next  balance, 
$800,  for  15  days  is  equivalent  to  the  interest  of  $12000  for  1  day,  etc.  The 
total  balance  of  interest  is  equivalent  to  the  interest  of  $112240  for  1  day,  or 
$18.71.  The  net  amount  due  is  $872  plus  $18.71,  or  $890.71.  (See  Art. 
446.) 

NOTES. — 1.  If,  at  any  time  in  the  above  operation,  there  had  been  a  credit 
balance,  it  would  have  been  necessary  to  have  had  additional  columns  for 
"Cr.  Balances"  and  "Cr.  Products." 

2.  The  above  method  is  used  by  bankers  and  trust  companies  that  pay 
interest  to  depositors  upon  their  "  daily  balances." 


EXAMPLES. 

594.     1.  Find   the   balance   due  on  the  following  account, 
Jan.  1,  1889;  interest  being  reckoned  at  6%. 


Dr. 


HOWARD  THORNTON. 


Cr. 


1888. 

1888. 

July    1 

Balance. 

1830 

45' 

Sept.  13 

Net  Proceeds. 

876 

40 

Aug.  24 

Mdse. 

448 

00 

Oct.  31 

a              <f 

912 

36 

Oct.  18 

Draft  C.  &  C. 

387 

40 

Nov.    5 

Cash. 

1000 

00 

Dec.  12 

Draft  H.  &  C. 

516 

88 

Art.  594.] 


ACCOUNTS     CURRENT. 


259 


2.  What  is  the  net  amount  due  on  the  following  account, 
July  1,  1882,  at  6%  ? 

Dr.  C.  H.  MILLS  in  %  current  with  G.  F.  SWORTFIGUER.   Cr. 


1881. 

1881. 

July     1 

Balance. 

1275 

46 

Nov.  14 

Mdse.    4  mo. 

587 

19 

Sept.  13 

Draft  #1012. 

871 

52 

1882. 

1882. 

Mar.  13 

"      30  da. 

612 

35 

Jan.      4 

"      #1017. 

913 

27 

Apr.   27 

"       60  da. 

846 

93 

May   17 

"      #1024. 

345 

63 

June    3 

Cash. 

500 

00 

3.  What  is  the  balance  of  the  following  account,  Apr.  1,  1882, 


at 


Dr.    W.  J.  HILLIS  in  account  with  LAKGRAVE  SHULTS.     Cr. 


1882. 

1882.       | 

Jan.  16 

Dft.  M.  &  C. 

937 

64 

Jan.     1 

Balance. 

3456 

75 

"      31 

«    B.  &  D. 

856 

75 

"      27 

Sales  as  Mar.  15 

1225 

19 

Mar.    3 

«     W.  &  Y. 

1749 

30 

Feb.    4 

Mdse  as  Mar.    6 

673 

75 

"     24 

«    V.  &0. 

912 

38 

"     28 

Sales  as  Mar.  19 

2428 

35 

4.  Find  the  amount  due  Aug.  1,  at  6$,  on  the  account  repre- 
sented in  Ex.  7,  Art.  574.     (See  Note,  Ex.  7,  Art.  574.) 

5.  Find  the  amount  due  Oct.  1,  1882,  at  6%,  on  the  account 
represented  in  Ex.  4,  Art  582. 

6.  Find  the  balance  due  Apr.  1,  1882,  at  6$,  on  the  following 
account  current. 

PHENIX  MILLS  in  %  current  with  JAMES  TALCOTT,  New  York, 
Apr.  1,  1882. 


Date. 

Dr. 

Amounts. 

Date. 

Cr. 

Amounts. 

1882. 

Mar.    1 

Balance. 

45108 

34 

1882. 

Mar.  31 

Net  Proceeds 

«    16 

Draft  #676. 

1000 

of  Account 

"    18 

"     #675. 

2000 

Sales  due  Apr. 

"    24 

"     #678. 

5000 

26,  1882. 

12505 

70 

«    28 

Cotton  Bill. 

3176 

42 

(See  Ex.  4, 

"    30 

Transportation. 

875 

10 

Art.  586.) 

260 


ACCOUNTS     CURRENT. 


[Art.  594. 


7.  Find  the  gain  or  loss  on  the  following  consignment  account, 
taking  as  the  day  of  settlement  Jan.  29,  1881,  the  day  the  draft 
for  the  balance  of  the  account  was  drawn  and  sold,  and  reckoning 
interest  at  6%  (365  days  to  the  year). 

Cons.  F.  L.  BEUCKMANN,  #14. 


1880. 

Dr. 

Days. 

Interest. 

Amounts. 

Apr. 

« 

25 
25 

Mdse.  Net  Cash.                                 ) 
Clearance.                                           i 

279 

300 

17 

(6544 

72 
20 

May 

10 

Insurance. 

*** 

# 

** 

40 

1881. 

Jan. 

29 

Balance  of  Interest  to  debit. 

#** 

•»# 

it 

29 

Gain. 

*** 

*# 

*** 

** 

**** 

## 

1880. 

Or. 

~ 

May 

7 

Draft  18000  Reichsmarks 

#•*•>;• 

*** 

•*•* 

4258 

42 

Nov. 

20 

"       2000 

#* 

* 

## 

468 

75 

1881. 

Jan. 

29 

"       9998 

0 

2368 

28 

« 

29 

Balance  of  Interest  to  debit. 

#*# 

*# 

*** 

*# 

#**# 

*# 

8.  What  was  the  amount  clue  on  the  following  account  Feb. 
13,  1881,  the  estimated  due  date  of  a  sight  draft  drawn  Jan.  29, 
1881,  for  the  balance,  reckoning  interest  at  5%  (365  days  to  the 
year) ? 

F.  L.  BRUCKMANN  on  account  of  Consignment  #14. 


1880. 

Dr. 

Days. 

Interest. 

Amounts. 

Oct. 

25 

Account  Sales                 due  Jan.  9,  1881 

35 

44 

80 

9344 

82 

Dec. 

31 

"            "                         "    Mar.  7,  1881 

22417 

M 

1881. 

Feb. 

13 

Balance  of  Interest  to  credit. 

*** 

** 

*** 

** 

***** 

** 

1880. 

Or. 

ifuTie 

80 

Freight                            due  May  14,  1880 

*** 

** 

** 

1176 

83 

May 

G 
G 

Draft  60  days'  sight        "    July  18,  1880  1 
"     69      "                        "       "     18,  1880  i 

*** 

**# 

** 

{    8COO 
<  10000 

Nov. 

19 

"     60     "          "           "    Feb.    1,1881 

** 

* 

** 

1881 

2000 

Feb. 

13 

Interest  Rm.  22417.54      "    Mar.    7,  1831 

** 

** 

** 

u 

13 

Balance  of  Interest  to  credit. 

*** 

** 

Jan. 

29 

Draft  at  sight  to  balance  due  Feb.  13,  1881 

**** 

** 

*** 

** 

***** 

** 

Art.  594.] 


ACCOUNTS     CURRENT. 


261 


NOTES.— 1.  The  interest  on  all  items  falling  due  after  the  day  of  settle- 
ment should  be  entered  in  the  interest  column  on  the  opposite  side  of  the 
account. 

Some  accountants  enter  these  items  of  interest  on  the  same  side  of  the 
account  in  red  ink  so  that  they  will  not  be  added  to  the  other  items,  and  transfer 
the  "  red  interest "  in  one  amount  to  the  opposite  side. 

2.  The  foregoing  represents  an  account  in  German  marks  (reichsmarks)  kept 
in  an  auxiliary  book  by  a  consignor  of  merchandise  to  a  commission  merchant 
at  Hamburg,  Germany. 

The  due  dates  of  drafts,  accounts  sales,  and  other  items  are  obtained 
from  the  letters  from  the  commission  merchant  and  from  accounts  sales  and 
memoranda  rendered  by  him.  The  correspondiDg  consignment  account  as 
entered  in  the  books  of  the  consignor  is  represented  in  Ex.  7. 


9.  What  was  the  balance  due  Jan.  1,  1882,  at  6$,  on  the 
account  represented  in  Ex.  5,  Art.  582. 

10.  Find  the  amount  due  Mar.  1,  1883,  at  §%,  on  the  account 
represented  in  Ex.  6,  Art.  582. 

11.  Calculate  the  interest  Jan.  1,  1883,  in  the  following  partial 
account  current,  and  find  the  total  amounts.     (Interest  6$,  365 
days  to  the  year.)     (See  Art.  587,  3.) 


G.  D.  SLOCUM  in  account  with  W.  B. 


1882. 

Dr. 

Days. 

Interest. 

Amounte. 

May 

1 

Totals  from  statement  of  May  1.  1882. 

1387 

63 

28765 

72 

« 

6 

Draft  H.  B.  Claflin  &  Co. 

240 

50 

71 

1285 

43 

« 

9 

"     Austin,  Nichols  &  Co. 

#** 

** 

** 

674 

89 

« 

13 

"    W.  H,  Schieffelin  &  Co. 

*** 

** 

** 

346 

27 

<t 

25 

"    Early  &  Lane. 

*** 

## 

** 

418 

43 

K 

28 

"    Mitchell,  Vance  &  Co. 

**# 

## 

** 

576 

80 

###* 

** 

***** 

** 

1882. 

Or. 

May 

1 

Totals  from  statement  of  May  1,  1882. 

973 

42 

22413 

71 

" 

5 

Sales  as  June  28,  1882. 

**# 

*7f* 

#* 

7316 

84 

tt 

12 

«      "  Aug.     1,  1882. 

*#* 

#* 

** 

2110 

92 

« 

18 

"      "  July  13,  1882. 

#** 

MM 

*«• 

13446 

85 

tt 

25 

Cash. 

*** 

** 

#* 

2000 

*#** 

** 

***** 

#•» 

262 


ACCOUNTS     CURRENT. 


[Art.  594. 


12.  Find  the  balance  due  on  the  following  account  Feb.  133 
1881.     (5$,  365  days  to  the  year.) 

Dr.      A.  WEI^GREEN  &  Co.,  on  account  of  Cons.  #25.       Cr. 


Date. 

Days. 

Interest 

Amounts. 

Date. 

Days. 

Interest. 

Amouii 

1880. 

1880. 

Pec. 

31 

Ace.  Sales 

Aug. 

2 

Freight. 

*** 

M 

** 

653 

due  Feb. 

Nov. 

19 

Draft  due  Feb.  1,1881. 

** 

M 

** 

18000 

19,  1881. 

22587 

89 

1881. 

Feb. 

13 

Interest  Rm.  22587.89. 

* 

** 

tt« 

1881. 

Feb. 

13 

Balance  of  Interest. 

** 

Feb. 

13 

Balance  of 

Jan. 

29 

Draft  to  balance  due 

Interett. 

M 

*»_ 

™ 

Feb.  13,  1881. 

— 

** 

**** 

13.  Find  the  net  gain  or  loss  on  the  following  consignment 
account,  Jan.  29,  1881.     (Interest  6%,  365  days  to  the  year.) 


Dr. 


Cons.,  A.  WEINGEEEN  &  Co.,  #25. 


Cr. 


Date. 

Days. 

Interest. 

Amounts. 

Date. 

Days. 

Interest 

Amoui 

1880. 

1880. 

June 

30 

Mdse. 

*** 

MM 

** 

4932 

86 

Nov. 

20 

Draft  Rm.  18000 

•» 

** 

*# 

4218 

July 

3 

Clearance. 

**» 

** 

20 

1881. 

Aug. 

1 

Insurance. 

»#* 

M 

25 

Jan. 

29 

"        "      3869 

0 

916 

1881. 

• 

" 

2-: 

Bal.  of  Interest. 

*** 

## 

Jan. 

29 

Bal.  of  Interest. 

*** 

«* 

29 

Gain. 

*** 

«* 



^ 

1L 

** 

**** 

14.  Find  the  amount  due  July  1,  1881,  on  the  account  repre- 
sented in  Ex.  7,  Art.  582. 

15.  What  was  the  balance  due  Jan.  1,  1883,  on  the  account 
represented  in  Ex.  8,  Art.  582  ? 

16.  Find  the  balance  of  the  following  account,  Mar.  31,  1882, 
at  6$. 

Dr.    JAMES  A.  DOUGLAS  in  %  current  with  J.  H.  HOYT.    Cr. 


1882. 

1882. 

Feb.  28 

Balance. 

18452 

50 

Mar.  8 

100  K  Y.C. 

14537 

50 

Mar.  2 

Draft. 

700 

"  11 

50H.&St.J. 

5162 

50 

«  11 

100  N.  W. 

14062 

50 

«  17 

Cash. 

16000 

"  18 

200H.&St.J. 

20875 

"  24 

100  K  W. 

14437 

50 

STOCKS    AND    BONDS. 


595.  "Stock"  is  a  term  applied  to  the   share  capital  of  a 
company,  and  represents  an  interest  in  its  property  over  and  above 
its  liabilities,  and  in  the  profits  of  its  business  after  the  expenses 
and  interest  on  its  bonds  have  been  paid. 

1.  A  Dividend  is  that  part  of  the  profits  of  a  company  which  is  divided 
among  the  stockholders,  and  is  a  certain  amount  per  share  or  a  certain  per 
cent,  of  the  par  value  of  the  stock. 

2.  The  Capital  Stock  of  a  company  is  divided  into  shares  usually  of  $100 
each.     Shares  of  $50  and  $25  are  called  half-stock  and  quarter-stock  respect- 
ively. 

3.  A  Stock  Certificate  is  a  written  instrument  issued  by  a  company,  and 
signed  by  the  proper  officers,  certifying  that  the  holder  is  the  owner  of  a  cer- 
tain number  of  shares  of  its  Capital  Stock. 

4.  The  Par  Value  is  the  sum  for  which  the   shares  or  certificates  were 
issued,  or  the  amount  mentioned  on  their  face.     The  Market  Value  is  the 
amount  for  which  they  can  be  sold. 

596.  A  Preferred  Stock  is  one  taking  preference  of  the 
ordinary  stock  of  a  corporation  in  the  payment  of  dividends. 

Thus,  the  holders  of  preferred  stock  of  a  certain  railroad  are  entitled  to 
6  per  cent,  on  their  stock  out  of  any  one  year's  earnings,  before  the  common 
stock  can  receive  any  dividend.  After  such  payment,  the  balance  of  earnings, 
if  any  remain,  may  be  divided  to  the  common  stock. 

Preferred  stocks  are  generally  the  result  of  a  reorganization  of  a  railroad. 
For  instance,  the  holders  of  the  common  stock  may  save  the  road  from  passing 
out  of  their  hands  by  the  payment  of  a  certain  sum  of  money,  for  which 
preferred  stock  is  issued.  In  other  cases,  preferred  stocks  have  been  issued 
in  payment  of  floating  or  unsecured  debts. 

In  some  reorganizations,  there  are  two  or  more  classes  of  preferred  stock. 

597.  A  Bond  is  the  obligation  of  a  Corporation,  City,  County, 
State,  or  Government  to  pay  a  certain  sum  of  money  at  a  certain 
time,  with  a  fixed  rate  of  interest  payable  at  certain  periods. 


2G4  STOCKS     AND     BONDS.  [Art.  597. 

1.  Bonds  of  business  corporations  are  usually  secured  by  a  mortgage  on 
the  whole  or  some  specified  portion  of  their  property. 

2.  Coupon  Bonds  are  those  with  small  certificates  attached  representing 
the  different  installments  of  interest  payable  at  the  different  periods  specified, 
during  the  time  the  bond  has  to  run,  which  are  to  be  cut  off  and  collected 
from  time  to  time  as  the  interest  becomes  due. 

3.  Bonds   are   also    issued   without   coupons,  in  what   is   known  as  the 
registered   form.      In  this  case  the  bond  is  only  payable  to  the  registered 
owner,  or  his  assignee,  and  the  interest  is  paid  by  check  or  in  cash,  to  the 
owner  or  his  attorney. 

4.  Bonds  are  sometimes  issued  with  coupons  attached  payable  to  bearer, 
but  the  principal  of  which  may  or  may  not  be  registered  at  the  choice  of  the 
owner. 

5.  Bonds  are  known  as  First  Mortgage,  Second  Mortgage,  etc.,  Consols, 
Sinking  Fund,  Income  or  otherwise,  according  to  their  priority  of  lien,  the 
class  of  property  upon  which  they  are   secured,   or  other  characteristics. 
Income  bonds  are  generally  bonds  on  which  the  interest  is  only  payable  if 
earned,  and  ordinarily  are  not  secured  by  a  mortgage. 

Bonds  are  also  named  from  the  rate  of  interest  they  bear,  or  from  the 
dates  at  which  they  are  payable  or  redeemable,  or  from  both;  as,  U.  S.  4's 
1907,  Virginia  6's,  Western  Union  7's,  coupon,  1900,  Lake  Shore  reg.  3d,  1903. 

6.  In  speaking  of  the  income  from  bonds  the  term  "  interest "  is  used,  as 
it  is  the  consideration  received  for  the  use  of  money  loaned, while  that  derived 
from  an  investment  in  stock  is  called  "dividend,"  because  it  is  money  divided 
to  the  stockholders  from  the  profit  of  carrying  on  the  business,  alter  the  fixed 
charges  have  all  been  paid. 

7.  Bonds  are  issued  in  denominations  of  $50  to  $50000. 


GOVERNMENT     BONDS. 

598.  4|  's  of  1 89 1 .    Redeemable  at  the  option  of  the  Govern- 
ment after  Sept.  1,  1891.     The  amount  outstanding  July  1, 1887, 
was  $250,000,000.     Interest  is  payable  Mar.  1,  June  1,  Sept.  1, 
and  Dec.  1. 

599.  4's  of  1907.    Redeemable  at  the  option  of  the  Govern- 
ment after  July  1,  1907.     The  amount  outstanding  July  1,  1887, 
was  $737,800,600.     Interest  is  payable  Jan.  1,  Apr.  1,  July  1,  and 
Oct.  1. 

GOO.  Refunding  Certificates.  These  certificates  are  of  the 
denomination  of  $10,  bear  interest  at  4$,  and  are  convertible  at 
any  time,  with  accrued  interest,  into  k.%  bonds.  The  amount 
outstanding  July  1,  1887,  was  $175,250. 


Art.  601.]  GOVERNMENT    BONDS.  265 

OO1.  Currency  6's.  These  bonds  were  issued  to  aid  in  the 
construction  of  the  Pacific  railroads.  Principal  and  interest  are 
payable  in  lawful  money  of  the  United  States.  Payable  30  years 
after  date,  and  maturing  at  different  dates  from  1895  to  1899.  The 
amount  outstanding  July  1,  1887,  was  $64,623,512,  all  registered. 

602.  Denominations.      The  coupon  bonds  of  the  various 
issues  are  in  denominations  of  $50,  $100,  $500,  and  $1000.     The 
registered  bonds  are  in  denominations  of  $50,  $100,  $500,  $1000, 
$5000,  and  $10000.     Of  the  4J's  of  1891,  and  the  4's  of  1907, 
there   are,   in   addition   to   the   above,   registered   bonds   of  the 
denominations  of  $20,000  and  $50,000. 

603.  Coupon  bonds,  being  payable  to  bearer,  pass  by  delivery 
without  assignment,  and  are  therefore  more  convenient  for  sale 
and  delivery  than  registered  bonds,  which  must  be  assigned  by  the 
party  in  whose  name  they  are  registered.     The  interest  coupons 
being  also  payable  to  the  bearer  will  be  cashed  by  any  bank  or 
banker  in  any  part  of  the  United  States. 

1.  The  interest  on  registered  bonds  is  paid  by  checks,  made  to  the  order 
of  the  registered  owner  and  sent  to  him  by  mail.     These  checks,  when  prop- 
erly endorsed,  can  be  collected  and  cashed  through  any  bank  or  banker. 

2.  Coupon  bonds   may  be  converted  into  registered  bonds  of  the  same 
issue,  but  there  is  no  provision  of  law  for  converting  registered  bonds  into 
coupon  bonds. 

604.  The  quotations  of  government  bonds  at  the  New  York 
Stock  Exchange  were  as  follows,  July  1,  1887  : 


Bid.       Asked. 


IT.  S.  4J's,  '91  reg.  109f  109f 

U.  S.  4}'s,  '91  c.  109}  109} 

U.  S.  4's,  1907  reg.  128£  128} 

U.  S.  4's,  1907  c.  128J  128} 

U.  S.  cur.  6's,  1895  123}  


Bid.   Asked. 


U.  S.  cur.  6's,  1896  126f 

U.  S.  cur.  6's5  1897  129} 

U.  S.  cur.  6's,  1898  132} 

U.  S.  cur.  6's,  1899  134} 
Dist.  of  Col.  3-65's 


All  Government  Bonds  are  dealt  in  and  quoted  "flat"—*',  e.,  the  quoted 
market  price  is  for  the  bond  as  it  stands  at  the  time,  including  the  accrued 
interest — except  that  after  the  closing  of  the  transfer  books  *  the  registered 
bonds  are  quoted  ex-interest — that  is  to  say,  the  interest  then  coming  due 
belongs  to  the  holder  of  the  bond  at  the  time  of  the  closing  of  the  books,  and 
does  not  go  with  the  bond  to  the  purchaser. 


*  The  transfer  books  of  U.  S.  registered  bonds  are  closed  for  the  month  preceding  the 
day  on  which  the  interest  is  paid. 


26G  STOCKS     AND     BONDS.  [Art.  604. 

During  tlit  period  in  which  the  transfer  books  remain  closed,  the  quoted 
price  of  coupon  bonds  includes  the  accrued  interest  falling  due  on  the  first 
of  the  ensuing  month,  while  that  of  registered  bonds  does  not.  If,  in  the 
month  of  December,  when  the  books  are  closed  preparatory  to  the  payment 
of  the  interest  due  January  1,  the  coupon  Four  per  cents  are  quoted  at  118, 
the  equivalent  for  the  registered  bonds  of  the  same  issue  would  be  117,  the 
three  months'  interest  being  equal  to  one  per  cent. 

NEW     YORK    STOCK     EXCHANGE. 

605.  The  New  York  Stock  Exchange  is  an  unincor- 
porated body  of  brokers,,  whose  business  is  to  buy  and  sell  stocks, 
bonds,  and  other  representatives  of  value. 

1.  The  floor  of  the  Exchange  is  open  for  business  from  10  A.  M.  to  3  p.  M. 
There  are  two  regular  calls  of  Stocks  daily  ;  three  of  State  and  Railroad 
Bonds ;  and  three  of  United  States  Bonds.     Transactions  are  not,  however, 
confined  to  the  regular  calls,  but  are  continually  taking  place  on  the  floor  of 
the  Exchange  between  the  hours  named  above. 

2.  In  Wall  Street,  there  are  what  are  known   as   strictly   commission 
houses,  who  take  and  execute  orders  for  securities,  charging  the  regular  com- 
mission, and,  when  customers  desire,  loaning  funds  on  the  securities  on  a 
deposit  of  10  to  20%  of  market  value  being  made.    This  is  what  is  known  as 
buying  on  a  margin  (GOO),  where  the  customer  intends  to  sell  soon  again, 
and  merely  buys  for  speculative  purposes.     Such  houses  will  usually  sell 
stocks  "  short "  (61O,  10)  for  their  customers  on  a  similar  margin. 

There  are  other  houses  which  make  no  advances,  and  require  customers 
to  pay  outright  for  securities  when  bought. 

There  are  also  houses  which  combine  a  banking  and  brokerage  business, 
taking  deposits  and  loaning  money  on  any  securities  marketable  at  the  Ex- 
change, and  buying  and  selling  stocks  on  commission.  Some  of  these  extend 
the  privilege  of  marginal  business  to  their  customers,  while  others  do  not. 

There  are  other  members  who  operate  exclusively  for  their  own  account. 

606.  Quotations  are  made  at  so  much  per  cent,  on  the  basis 
of  a  par  value  of  $100  per  share  of  stock,  except  in  the  case  of 
mining  securities  and  Sutro  Tunnel  stock,  which  are  quoted  at  so 
much  per  share,  without  reference  to  their  par  value. 

For  example,  the  par  value  of  Morris  and  Essex  stock  is  $50,  but  the 
quotation,  if  the  stock  were  worth  just  par  in  the  market,  would  be  100 %  ;  or, 
if  the  quotation  is  110,  it  means  $110  for  $100  worth  of  the  par  value,  which, 
in  the  case  of  this  stock,  would  be  two  shares,  while  in  the  case  of  a  stock  the 
par  value  of  which  is  $100  per  share,  it  would  be  for  one  share. 

On  the  other  hand,  if  Sutro  Tunnel,  the  par  value  of  which  is  $10  per 
share,  is  quoted  at  2,  it  means  $2  per  share  ;  and,  in  like  manner,  if  Home- 
stake,  the  par  value  of  which  is  $100,  is  quoted  at  30,  it  means  $30  per  share. 


Art,  607.]         NEW     YORK     STOCK     EXCHANGE.  267 

607.  Commission. — The  regular  charge  for  buying  and 
selling  securities  dealt  in  at  the  Stock  Exchange,  except  mining 
stocks,  is  one-eighth  of  one  per  cent.  (\%)  on  par  value,  or  $12.50 
on  100  shares  of  stock  of  the  par  value  of  $100  each. 

608.  Stocks    are   usually  bought    or    sold    either   "cash," 
"regular  way/'  "  seller  three,"  or  "buyer  three."    A  stock  sold 
"  cash  "  is  deliverable  the  day  sold  ;  a  stock  sold  "regular  way  " 
is  deliverable  the  next  day,  or  bought   "  regular  way  "  is  to  be 
paid  for  the  next  day.     Where  nothing  else  is  specified,  "regular 
way  "  is  always  understood.     When  a  stock  is  reported  as  bought 
"seller  three,"  it  is  meant  that  the  seller  of  the  stock  can  deliver 
it  on  either  of  the  three  days  at  his  option,  but  is  not  required  to 
deliver  until  the  third  day.     On  the  other  hand,  when  a  trans- 
action is  made  "  buyer  three,"  the  buyer  can  demand  delivery  of 
the  stock  at  any  time  within  three  days,  but  must  take  it  and  pay 
for  it  by  the  third  day. 

1.  Transactions  on  any  of  the  above  terms  carry  no  interest. 

2.  If  the  option  is  over  three  days,  six  per  cent,  on  the  selling  value  of 
the  stock  is  paid  by  buyer  to  seller. 

3.  One  day's  notice  is  required  of  intention  to  terminate  an  option  of  a 
longer  period  than  three  days. 

4.  The  Stock  Exchange  does  not  recognize  any  contract  for  over  sixty 
days.    Should  a  stock  pay  a  dividend  during  the  pendency  of  a  contract,  the 
dividend  belongs  to  the  purchaser  of  the  stock,  unless  otherwise  previously 
agreed. 

609.  A  Margin  is  a  deposit  made  with  a  broker,  by  a  person 
who  wishes  to  buy  or  sell  stock  for  speculation  to   enable   the 
broker  to  "carry"  the  stock  and  protect  himself  against  loss.     It 
is  usually  10%  of  the  par  value  of  the  stock. 

1.  A  person  desiring  to  speculate  in  stocks,  deposits  with  his  broker  $1000 
as  a  margin,  and  directs  him  to  purchase  100  shares  of  a  certain  stock  at  90. 
The  broker  would  pay  for  the  stock  $9000,  $1000  of  which  being  furnished 
by  the  speculator,   and  the  remainder,   $8000,  by  the  broker.     The  broker 
charges  legal  interest  on  the  amount  furnished  by  him  for  "carrying"  the 
stock.     (See  Ex.  52,  Art.  611.) 

2.  The  margin  deposited  with  the  broker  is  simply  to  protect  the  broker 
^against  losing  any  money  should  the  stock  move  in  the  wrong  direction.     In 

case  of  its  so  doing,  the  margin  must  be  made  good  by  the  deposit  of  an 
additional  amount,  otherwise  the  broker  will  sell  the  stock  to  protect  himself 
from  losing  any  of  the  money  he  has  advanced. 


268  STOCKS     AND     BONDS.  [Art.  61 0. 

61O.  Explanation  of  Words  and  Phrases  used  in  Wall  Street. 

1.  Bear.  An  operator  who  is  "short"  of  stock.    He  wishes  to  buy  at  a 
lower  rate,  and  tries  to  depress  the  price  of  the  stock  of  which  he  is  "  short." 

2.  Bull.  An  operator  who  is  holding  stock  for  an  advance.     He  is  said  to 
be  "long  "of  the  stock.     Bulls  try  to  advance  the  prices  of  the  stocks  of 
which  they  are  "long." 

3.  b.  3  (Buyer  3),  10,  20,  30,  etc.     Meaning  at  the  buyer's  option,  within 
three  days,  ten  days,  etc.     "When  in  a  stock  transaction,  the  buyer  has  the 
privilege  of  taking  the  stock  at  any  time  during  the  number  of  days  mentioned. 
In  buyer's  options,  when  the  option  is  for  more  than  three  days,  six  per  cent, 
interest  is  charged  the  buyer,  and  the  seller  is  entitled  to  one  day's  notice. 

4.  b.  c.,  "between  calls."    The  sale  not  taking  place  on  the  call  of  the 
stock,  but  after  the  first  call  and  before  the  second. 

5.  Collaterals.     Stocks,  bonds,  notes,  or  other  value  given  in  pledge  as 
security,  when  money  is  borrowed. 

6.  Cover,  to  "  cover  one's  shorts."    Where  stock  has  been  sold  short,  and 
the  seller  buys  it  in  to  realize  his  profit,  or  to  protect  himself  from  loss,  or  to 
make  his  delivery.     This  is  "covering  short  sales." 

7.  Differences.    When  the  price  at  which  a  stock  is  bargained  for  and  the 
rate  on  day  of  delivery  are  not  the  same,  the  broker  against  whom  the  varia- 
tion exists,  frequently  pays  the  "difference"  in  money,  instead  of  furnishing 
or  receiving  the  stock. 

8.  Ex.-Div.,  Ex.-Dividend.     When  the  price  of  a  stock  does  not  include, 
and  the  stock  does  not  carry  to  the  buyer  a  recently  declared  dividend. 

9.  Seller,  3,  10,  20,  80,  etc.     Sold  deliverable  at  seller's  option,  within  the 
number  of  days  named.     When  seller's  options  are  for  more  than  three  days, 
the  buyer  pays  six  per  cent,  interest,  unless  "  flat "  is  specified  in  the  contract, 
and  the  seller  must  give  one  day's  notice  of  delivery. 

10.  Short.     When  one  has  sold  stock  which  he  does  not  own,  hoping  to 
realize  a  profit  by  buying  in  at  lower  prices,  he  is  said  to  be  "  short." 

11.  Watering  a  Stock.      The   act  of  increasing  the  quantity  of  a  stock 
without  a  corresponding  increase  in  the  value  of  the  property  which  it  repre- 
sents.    This  is  usually  done  in  the  reorganization  of  a  railroad,  or  in  the  con- 
solidation of  two  or  more  railroads. 


EXAM  PLES. 

611.  1.  A  bank  with  a  capital  (595)  of  $250,000,  declares 
a  semi-annual  dividend  of  3|%.  What  is  the  amount  of  the  divi- 
dend, and  how  much  will  a  stockholder  receive  who  owns  16  shares 
of  $100  each  (595,  2)  ? 

2.  An  insurance  company  divides  among  its  stockholders 
$18000.  What  is  the  rate  of  the  dividend,  the  capital  stock  being 
$225000  ?  How  much  is  paid  to  Mr.  A.,  who  has  a  certificate 
(595,  3)  for  25  shares  ? 


Art.  611.]  STOCKS     AND     BONDS.  269 

8.  A  gas  company  declares  a  dividend  of  5$,  and  divides  among 
its  stockholders  $125000.     What  is  its  capital  stock  ? 

4.  The  board  of  directors  of  a  mining  company  declared  a  divi- 
dend of  $100,000,  being  five  cents  per  share  (par  value  $10)  on 
the  capital  stock  of  the  company.     What  was  the  capital  stock, 
and  in  how  many  shares  was  it  divided  ?    The  dividend  was  what 
per  cent,  of  the  capital  stock  ? 

5.  An  installment  of  10$  was  assessed  and  called  on  the  capital 
stock  of  a  new  railroad  company.     How  much  was  paid  by  Mr.  B. , 
who  had  subscribed  for  50  shares  (par  value  $100)  ? 

6.  A   railway   company,   whose   capital   stock   is   $1,750,000, 
declares  a  dividend  of  34-  per  cent.     What  was  the  amount  of  the 
dividend  ? 

7.  The  Union  Pacific  Railway  paid  to  its  stockholders,  in  1879, 
$2,204,700.     What  was  the  par  value  of  its  stock,  the  rate  of  the 
dividend  being  6%  ? 

8.  A  quarterly  dividend  of  §\%  was  declared  by  a  manufactur- 
ing company.     What  was  the  capital  stock,  the  amount  of  the 
dividend  being  $2100  ? 

9.  What  is  the  market  value  of  200  shares  of  stock  at  116|  ? 

10.  How  many  shares  of  W.  II.  Tel.  can  be  bought  for  $43725 
at  79J-$  ? 

11.  What  is  the  total  par  value  (595,  4)  and  the  total  market 
value  of  100  shares  Lake  Shore  at   118|  (606),  300  sh.   N.  J. 
Central  at  89},  500  sh.  W.  U.  Telegraph  at  78£,  200  sh.  U.  S. 
Express   at   73J,  500  sh.   N.  Y.,  L.  E.  &  W.   com.  at  40£,  and 
800  sh.  1ST.  Y.,  L.  E.  &  W.  pref.  (596)  at  90|  ? 

12.  What  is  the  cost  of  250  shares  Tex.  &  Pac.  at  50|  and  100 
shares  Ohio  &  Miss.  pref.  at  104,  brokerage  \%  (6O7)  ? 

13.  What  are  the  proceeds  of  600  shares  Morris  and  Essex  (half 
stock,  595,  2)  sold  through  a  broker  at  121|  ? 

14.  What  are  the  proceeds  of  the  following  stocks  sold  through 
a  broker?  200  shares  Union  Pacific  at  117-f,  2000  shares  N.  Y., 
0.  &  W.  at  27J-,  800  shares  A.  &  T.  H.  pref.  at  88,  and  600  shares 
Chi.  &  Alton  at  131J. 

15.  Find  the  cost  of  10  shares  Manhattan  Bank  at  135,  $5000 
Erie  7's  (597,  5)  cons,  gold  bonds  (597)  at  128,  $1000  Toledo 
and  Wabash  2d  (597,  5)  at  108 J,  $5000  C.  E.  I.  &  P.  6's,  1907, 
coupon  (597,  2)  at  129,  and  $5000  Ohio  Southern  Income  (597,  5) 
at  45,  usual  brokerage. 


270  STOCKS     AND     BONDS.  [Art.  611. 

16.  Find  the  proceeds  of  $15000  U.  S.  4's,  registered,  1907 
(599),  b.  3,  at  117£,  and  $10000  U.   S.  4f  s  coupon  (598)  at 
114J,  usual  brokerage. 

17.  How  much  must  be  invested  in  U.  S.  4J's,  1891,  to  pro- 
duce a  quarterly  income  of  $675,  bonds  selling  at  114|  ? 

18.  When  Ohio  6's  are  sold  at  109J,  what  is  received  for  six 
$500  bonds,  brokerage  ±%  ? 

19.  When  Pittsburg,  Fort  Wayne  and  Chicago  2d  7's,  1912, 
are  worth  135,  what  will  $12000  in  bonds  cost  ? 

20.  How  many  $500  bonds  shall  I  receive  for  $4735  invested 
in  U.  S.  4's  at  118f  ? 

21.  How  much  must  be  sent  to  a  broker  that  he  may  purchase 
$8000  IT.  S.  4i's  at  102J,  commission  \%  ? 

22.  An  executor  sold  Central  of  New  Jersey  stock  at  52|,  and 
purchased  with  the  proceeds  $42000  in  U.  S.  4's,  1907,  at  lOOf . 
What  was  the  par  value  of  the  stock  sold,  usual  brokerage  ? 

28.  A  broker  bought  on  his  own  account  200  sh.  Nor.  Pac.  pf. 
at  69-J,  and  sold  the  same  the  same  day  at  73|.  What  was  his 
gain  ? 

24.  How  many  shares  of  111.  Cen.  bought  at  129|,  and  sold  at 
132f ,  usual  brokerage,  will  produce  a  gain  of  $1375  ? 

25.  What  income  will  be  produced  by  investing  $235250  in  4% 
bonds  at  117|  ? 

26.  The  common  stock  of  a  railroad  company  is  $46,000,000, 
and  the   preferred   stock  (596)  $8,000,000.     The  company  de- 
clares a  dividend  of  3^%  on  the  preferred  stock  and  2^  on  the 
common  stock.     What   is   the  surplus,  if  the  net  earnings  are 
$1,317,645  ? 

27.  Bought  June  4,  800  sh.  Ohio  &  Miss.  pref.  at  35£,  s.  30. 
The  stock  was  delivered  June  24.     What  was  the  amount  paid 
including  interest  (61O,  9)  ? 

28.  Bought  May  16,  200  sh.  Lake  Shore  at  116|,  b.  60,  and 
called  for  the  stock  July  5.    What  was  the  cost  including  interest  ? 

29.  Jan.  10,  sold  100  sh.  Phil.  &  Read,  at  65J,  s.  3.     Jan.  13 
the  stock  was   quoted   at  68J.     How   much   was  the   difference 
(61O,  7)  paid  by  the  seller  in  settlement  ? 

30.  How  much  must  one  invest  in  ±\%  bonds,  when  they  are 
selling  at  121J,  in  order  to  have  an  income  of  $1800  per  year  ? 

SI.  A  man  invests  $28020  in  4^  stock  at  116$,  brokerage  \%. 
What  is  his  income  ? 


Art.  611.]  STOCKS     AND     BONDS.  271 

82.  I  sell  200  sh.  H.  &  St.  J.  pf.  at  lllf,  and  $10000  N.  Y. 
Elevated  1st  mortgage  bonds  at  119.     What  will  be  the  net  pro- 
ceeds of  the  sale,  allowing  usual  brokerage  ? 

83.  A  person  sells  26  shares  §\%  stock  at  86,  and  loans  the 
proceeds  at  b%.     What  is  the  increase  in  his  income  ? 

84.  July  1,   1887,   the   interest-bearing   debt   of  the  United 
States  was  as  follows:  4£'s,  $250,000,000;  4's,  $737,975,850;  3's 
$14,000,000 ;  Pacific  R,  R.   6's,  $64,623,512.     What  is  the  total 
interest  for  one  year  ? 

85.  The  capital  stock  of  a  railroad  company  was  "watered" 
(61O,  11)  by  declaring  a  stock  dividend  of  10%.     If  the  market 
value  of  the  old  stock  was  110,  what  should  be  the  value  of  the 
new  stock  ? 

86.  Jan.  1,  1882,  the  A.  &  B.  R.  R.,  having  a  capital  stock  of 
$20,000,000,  was  consolidated  with  the  B.  &  C.  R.  R.,  having  a 
capital  stock  of  $32,000,000.     The  new  company  was  organized 
under  the  name  of  the  A.,  B.,  &  C.  R.  R.     For  every  share  of  the 
A.  &  B.  R.  R.  there  was  issued  1|  shares  of  the  new  stock,  and  for 
every  share  of  the  B.  &  C.  R.  R.  there  was  issued  1|  shares  of  the 
new  stock.     What  was  the  capital  stock  of  the  new  company,  and 
how  much  was  the  stock  (e  watered"  ? 

37.  Before  the  consolidation,  the  stock  of  the  A.  &  B.  R.  R. 
was  worth  1.20  in  the  market,  and  the  stock  of  the  B.  &  C.  R.  R. 
90.  What  should  be  the  quotation  of  the  new  stock  ? 

88.  During  the  year  1881,  the  A.  &  B.  R.  R.  divided  among 
its  stockholders  $1,600,000,  and  the  B.  &  C.  R,  R.,  $1,920,000. 
During  the  year  1882,  the  new  company  divided  an  amount  equal 
to  the  total  dividends  of  the  two  companies  in  the  preceding 
year.  What  were  the  rates  of  the  dividends  of  the  two  companies 
in  1881,  and  the  rate  of  the  dividend  of  the  consolidated  company 
in  1882  ? 

39.  Mr.  A.  had  10  shares  of  the  A.  &  B.  R.  R.,  and  16  shares 
of  the  B.  &  C.  R.  R.     What  was  the  total  amount  of  his  dividend 
in  1881  ?     How  many  shares  of  the  new  stock  did  he  receive,  and 
what  was  the  amount  of  his  dividend  in  1882  ? 

40.  The  gross  earnings  of  the  M.  C.  R.  R.  for  the  year  ended 
Dec.  31,  1880,  were   $9,085,749;  operating  expenses  and  taxes, 
$5,738,751 ;  interest  and  rentals,  $1,586,410.     After  declaring  a 
dividend,  there  was  a  surplus  of  $261,532.     What  was  the  rate  of 
the  dividend,  if  the  amount  of  the  stock  was  $18,738,200  ? 


272  STOCKS     AND     BONDS.  [Art.  Gil. 

41.  A  gentleman  bought  bank  stock,  paying  regular  annual 
dividends  of  6$,  at  120.     What  was  the  rate  per  cent,  of  his  in- 
come, or  what  per  cent,  did  he  receive  on  the  money  invested  ? 

ANALYSIS. — Since  dividends  are  reckoned  on  the  par  value  of  the  stock, 
the  dividend  on  1  share  of  $100  would  be  $6.  Since  each  share  costs  $120,  and 
pays  $6  income,  the  per  cent,  would  be  $6-*-$120,  or  5%. 

NOTE. — The  above  analysis  will  not  apply  to  bonds  that  mature  at  a  cer- 
tain fixed  time,  unless  the  investor  expects  to  sell  the  bonds  before  maturity 
at  the  cost  price.  If  $%  bonds  that  mature  in  1899  are  purchased  in  1889  at 
120,  and  are  sold  at  the  same  rate  before  maturity,  they  will  pay  §%  on  the 
investment,  or  cost.  If  the  bonds  are  held  until  maturity  (1899),  or  for  10 
years,  the  owner  would  receive  from  the  government  the  par  value  only,  or 
$100  for  a  bond  of  that  amount,  and  the  bonds  would  yield  less  than  5%. 
If  6%  bonds,  maturing  in  10  years,  are  purchased  at  1.07^  and  held  until 
maturity,  they  will  pay  5%  on  the  investment.  (See  Ex.  59.)  If  Qfc  bonds, 
that  mature  in  2  years,  are  purchased  at  more  than  112,  there  would  be  a  loss 
of  interest  to  the  purchaser  instead  of  a  gain. 

42.  Which  is  the  better  investment,  stock  paying  a  regular 
annual  dividend  of   5$  and  bought  at  80,   or  stock  paying  8$ 
dividends,  and  bought  at  120  ? 

48.  If  insurance  stock  paying  regular  dividends  of  10$  annually 
is  bought  at  13  7J-,  brokerage  \%.  what  per  cent,  of  income  will  it 
produce  ? 

44-  Which  investment  will  produce  the  greater  annual  income 
and  how  much,  $20,000  invested  in  Chemical  Bank  stock  at  2000 
which  pays  dividends  of  15$  every  two  months,  or  the  same  amount 
invested  in  Chatham  Bank  stock  at  125  which  pays  regular  semi- 
annual dividends  of  3$  ? 

45.  What  rate  can  you  afford  to  pay  for  stock  paying  regular 
annual  dividends  of  10$,  in  order  to  realize  6$  on  the  invest- 
ment ? 

46.  At  what  price  must  8$  stocks  be  purchased  to  afford  5$  on 
the  investment  ?     To  afford  6$  ? 

47.  Stocks  bought  at  80  pay  regular  dividends  of  5%.     What 
is  the  rate  per  cent,  on  the  investment  ?    At  what  rate  should 
they  be  purchased  to  afford  4$  on  the  investment  ?    To  afford  8$  ? 

48.  Purchased  400  shares  Lake  Shore  at  118 J,  and  200  shares 
Chesapeake  and  Ohio  2d  pref.,  at  24|.     Sold  the  Lake  Shore  at 
113|,  and  the  Chesapeake  and  Ohio  at  22J.     What  was  the  loss, 
usual  brokerage,  no  interest  ? 


Art.  611.] 


STOCKS    AND     BONDS. 


273 


49.  The  gross  earnings  of  the  Union  Pacific  Railway  Co.  for 
1879,  were  $13,201,077.66  ;  the  operating  expenses  were  $5,475,- 
503.44.     What  were  the  surplus  earnings,  and  what  per  cent,  of 
the  gross  earnings  were  the  operating  expenses  ? 

50.  A  synopsis  of  the  report  of  the  N.  Y.  C.  &  H.  R.  R.  R.  for 
its  fiscal  year  ended  Sept.  30,  1881,  is  as  follows  :  Gross  earnings 
from  passengers,   $6,958,038;   from  freight,  $20,736,749;   from 
miscellaneous,  $4,653,608;  expenses,  $19,464,786;  interest,  rentals, 
and  taxes,  $4,990,783.    What  was  the  surplus  for  the  year  after  the 
declaration  of  a  dividend  of  8%  on  a  capital  stock  of  $89,229,300  ? 
The  expenses  were  what  per  cent,  of  the  total  earnings  ? 

51.  The  L.  S.  &  M.  S.  Railway  reported  as  follows  for  the  year 
ended    Dec.  31,  1880:    Gross   earnings,  $18,749,461;   operating 
expenses  and  taxes,  $10,418,105  ;   interest,  rentals,  dividend  on 
guaranteed  stock,  and  $250,000  for  the  sinking  fund,  $3,000,374. 
After  paying  a  dividend  of  8$,  there  was  a  surplus  for  the  year  of 
$1,373,662.     What  was  the  amount  of  the  dividend,  and  the  capi- 
tal stock  ? 

52.  July  26,  a  broker  received  from  a  customer  a  remittance 
of  $1000  as  a  margin  (6O9)  and  purchased  for  him  100  shares  of 
St.  Paul  Common  at  59.     On  Aug.  2,  the  broker  sold  the  stock  at 
64J.     What  was  the  customer's  profit  ? 


OPERATION. 


Dr. 

July  26. 

To  100  shares  St.  Paul  Com.  59 
Commission  \% 

.  $5900 
12.50 

5912 

50 

Aug.  2. 

Interest  $5912.50,  7  days      . 
Cr. 

• 

* 

** 

**** 

** 

July  26. 
Aug.  2. 

By  margin  deposited     . 
"  100  shares  St.  Paul  Com.  64£ 
Commission  \<fo 

.  $6450 
12.50 

1000 
6437 

50 

Aug.  2. 

Interest  $1000,  7  days  . 
Balance  . 

•      '  • 

* 

** 

#*#* 

** 

#*** 

** 

The  profit  is  equal  to  the  balance  less  $1000,  the  original  deposit. 

53.  Aug.  30,  a  broker  purchased  for  the  account  of  a  customer 
300  shares  Northwestern  Railroad  stock  at  78.  He  deposited  as 
a  margin  $3000.  On  Sept.  22,  the  stock  was  sold  at  74f.  What 
was  the  loss  ?  (Interest  6$,  usual  commission.) 


274  STOCKS     AND     BONDS.  [Art.  611. 

54.  May  10,  a  speculator  deposited  with  his  broker  $5000  as  a 
margin,  and  directed  him  to  purchase  for  his  account  500  shares 
N.  Y.,  L.  E.,  &  W.,  pref.  at  90f.     May  20,  the  stock  was  sold  at 
94J.     What  was  the  gain,  interest  6%,  usual  brokerage  ? 

55.  Sept.  10,  I  deposited  with  my  broker  $5000  as  a  margin, 
and  he  purchased  for  me  200  sli.  Cen.  Pac.  at  90J-,  200  sh.  Morris 
&  Essex  (half  stock)  at  122^,  200  sh.   Tex.  &  Pac.  at  49J.     The 
stocks  on  Sept.  30  were  quoted  as  follows  :  Cen.  Pac.  80J,  Morris 
&  Essex  120|,  Tex.  &  Pac.    41f .      How   much   should   I   have 
deposited  with  my  broker  to  make  my  margin  of  10%  good,  and 
to  cover  commission  for  buying  and  selling,  and  interest?     If  I 
had  been  unable  to  have  made  an  additional  deposit,  and  the  broker 
had  "  sold  me  out/'  what  would  have  been  my  loss  ? 

56.  An   operator,    supposing   Erie    would    decline   in   value, 
ordered  his  broker  to  sell  short    100   shares   at   50,    and  at  the 
same  time  deposited  with  him  as  a  margin  $1000.     The  broker 
on  receiving  the  order  sold  for  his  account  100  shares  at  50,  and 
borrowed  the  stock  for  delivery.     When  the  market  price  declined 
to  45,  he  ordered  the  broker  to  "cover  his  short  sale"  (buy  the 
stock  for  delivery),  and  return  the  stock  to  the  party  from  whom 
it  was  borrowed.     What  was  the  gain,  usual  brokerage  ? 

OPERATION. 

Or. 

By  margin  deposited $****.** 

"  100  shares  Erie  borrowed  and  sold  at  50.     .        .  ******        $****.** 

Dr. 

To  100  shares  Erie  bought  and  returned  at  45  .        $****.** 

"  commission  for  selling  the  stock  \%    .         .        .  **.** 

"  "  "    buying  and  returning  the  stock  \  % .        **.**          ****** 

"  amount  {o  credit.        .........        ****.** 

The  net  profit  equals  the  balance  less  the  margin  deposited. 

NOTE. — There  is  no  interest  charged  on  short  sales,    but   it   sometimes 
happens  that  a  small  bonus  has  to  be  paid  for  the  use  of  the  borrowed  stock. 

57.  A  broker  sold  "short"  for  me  400   sh.  C.  B.  &  Q.,  at 
135|,  and  100  sh.  C.  E.  I.  &  P.,  at  132 J.     My  "short"  sale  on 
C.  B.  &  Q.  was  "covered"  at  131 J,  and  C.  E.  I.  &  P.  at  133f. 
What  was  my  net  profit,  usual  brokerage  ?     (No  interest. ) 

58.  Sold  Aug.  11,  500  shares  Chicago  &  Alton,  s.  3,  at  94J, 
and  covered  my  short  sale  Aug.  14,  at  91.     What  was  my  profit, 
allowing  the  usual  brokerage  ? 


•Art.Gll.J  STOCKS    AND     PONDS.  275 

59.  At  what  price  may  6$  bonds,  maturing  in  10  years,  be 
purchased,  so  that  the  investment  will  pay  5%  ? 

NOTE. — Tables  have  been  constructed  on  various  plans,  and  different 
methods  are  used  by  bankers  and  financiers,  for  the  solution  of  problems 
relating  to  bond  investments ;  two  of  which  are  given  below. 

ANALYSIS. — 1.  In  the  following  method,  it  is  presumed  that  the  accruing 
interest  is  not  reinvested,  but  that  a  sufficient  part  of  it  is  set  aside  as  a  sink- 
ing fund  to  make  up  the  amount  which  was  originally  paid  out  as  premium. 

A  $1000  bond  in  10  years  at  Q%  would  amount  to  $1600  ($1000  + 10  x  $60). 
$1  in  10  years  at  5^  would  amount  to  $1.50.  To  amount  to  $1600,  the  prin- 
cipal, or  the  amount  paid  for  the  bond,  must  be  as  many  times  $1  as  $1.50  are 
contained  times  in  $1600,  or  $1066. 66|  (106f$). 

If  a  $1000  bond  is  purchased  at  106f,  it  will  be  necessary  to  set  aside  as  a 
sinking  fund  each  year  $6.66f  (\%}  to  make  up  the  premium  in  10  years. 
The  annual  interest,  $60,  less  $6|,  the  annual  sinking  fund,  is  $53^-,  which  is 
5%  of  $1066|,  the  cost  of  the  bond  or  the  amount  invested. 

If  the  amount  set  aside  as  a  sinking  fund  is  placed  at  interest,  either 
simple  or  compound,  6%  bonds,  maturing  in  10  years  and  purchased  at  106|, 
would  pay  a  little  more  than  §%. 

2.  The  following  method  anticipates  compound  interest  throughout;  i.  e., 
the  interest  is  immediately  reinvested  at  compound  interest. 

The  holder  of  a  $1000  bond  would  receive  $60  interest  annually,  and 
$1000,  the  face  of  the  bond,  in  10  years.  If  money  is  Worth  5%,  the  several 
interests  in  the  10  years  at  compound  interest  would  amount  to  $754.674 
($1  placed  at  compound  interest  at  the  beginning  of  each  year  would  amount 
in  9  years  to  $11.5779  (484).  $11.5779  plus  $1  of  the  last  interest  =  $12.5779. 
$60  would  amount  to  60  times  $12.5779,  or  $754.674).  $1000,  the  principal, 
plus  $754.674,  the  compound  amount  of  the  interest,  equals  $1754.674,  the 
•  total  value  of  the  bond  at  maturity,  money  being  worth  §%.  The  present 
worth  of  $1754.64,  due  in  10  years,  at  5%  compound  interest,  is  $1754.64+- 
$1.6289  (483),  or  $1077.19.  Hence  the  bonds  must  be  purchased  at  1.07T7«& 
to  pay  §%  on  the  investment.  (See  Ex.  41.  Note.) 

60.  What  must  I  pay  for  $%  bonds,   maturing  in  15  years, 
that  my  investment  may  yield  ^\%  ?     (Both  methods.) 

61.  6%  bonds,  maturing  in  10  years  and  bought  at  lOGf ,  pay 
what  per  cent,  on  the  investment  ?     (See  1st  analysis,  Ex.  59.) 

ANALYSIS.— A  $1000  bond  would  amount  in  10  years  at  6^  to  $1600.  If 
$1066.66f  is  paid  for  the  bond,  the  net  interest  for  10  years  is  $1600— $1066.66|, 
or  $533.33^;  and  for  one  year  $533.33i-f-10,  or  $53.33^.  An  income  of  $53.33^ 
on  an  investment  of  $1066.66|  is  equivalent  to  5^  ($53.33^  -r-  $1066. 66f). 

62.  What  rate  of  interest  do  I  receive  on  my  investment,  if  I 
buy  1%  bonds  maturing  in  20  years  at  133-J-  ? 


TAXES. 


612.  A  Tax  is  a    sum   of   money  assessed  on  persons  and 
property  to  defray  the  expenses  of  a  State,  county,  town,  corpo- 
ration, or  district. 

1.  In  certain  States  all  citizens  above  21  years  of  age  are  required  by  law 
to  pay  a  certain  tax  on  the  person.    This  tax  is  called  a  Capitation  or  Poll 
Tax. 

2.  The  expenses  of  States,  counties,  towns,  etc.,  are  paid  by  a  direct  tax 
upon  the  property  or  polls  of  the  same.     The  methods  of  assessing  taxes  differ 
in  the  several  States.     In  some  States,  a  certain  percentage  of  the  whole  tax  is 
assessed  upon  the  polls,  while  in  others  the  poll  tax  is  a  fixed  amount  for  each 
citizen.     In  certain  States,  the  whole  tax  is  paid  by  the  owners  of  the  property 
of  the  same.     In  the  States  of  New  York,  Pennsylvania,   and  some  other 
States,  there  is  a  direct  tax  levied  upon  certain  corporations  doing  business  in 
the  State. 

3.  The  expenses  of  the  United  States  government  are  paid  by  duties  on 
imports ;  the  internal  revenue  (the  tax  upon  distilled  spirits,  fermented  liquors, 
tobacco,  snuff,  and  cigars) ;  sales  of  public  lands ;  tax  on  circulation  of  national 
banks;  customs  fees,  fines,  penalties,  and  forfeitures;  fees,  consular,  letters 
patent,  and  land ;  profits  on  coinage,  etc. 

613.  Heal  Estate  is  fixed  property;  as  land,  houses,  etc. 

614.  Personal  Property  is  movable  property,  as   money, 
stocks,  bonds,  mortgages,  furniture,  merchandise,  etc. 

615.  An  Assessor  is  a  person  appointed  or  elected  to  esti- 
mate the  valuation  of  all  property  liable  to  taxation. 

616.  A  Collector  or  Receiver  of  taxes  is  a  person  appointed 
or  elected  to  collect  or  receive  the  taxes  of  a  city,  town,  village,  or 
district. 

Collectors  receive  a  commission  on  the  amount  collected  or  a  fixed  salary. 

EXAMPLES. 

617.  1.  For  the  fiscal  year  1879,  the  N.  Y.  State  tax  levy  was 
at  the  rate  2^%^  mills.     How  much  would  this  rate  produce,  the 
valuation  of  the  taxable  property  being  $2,686,140,000  ? 


Art.  61 7.] 


TAXES. 


377 


2.  The  rate  of  taxation  of  a  certain  county  was  3J  mills,  and 
the  amount  of  the  tax  $40,653.48.     What  was  the  valuation  of  the 
property  ? 

3.  The  following  were  the  rates  of  taxation  of  New  York  for 
State  purposes,  1880: — schools,   1.085  mills;  general  purposes, 
1.475  mills;  newcapitol,  .6  mills;  canals,  .34  mills.     What  was  the 
total  rate  of  taxation,  and  how  much  was  raised  by  a  county  whose 
valuation  was  fixed  by  the  State  Board  of  Equalization  at  $11,047,- 
534  ?     How  much  was  raised  for  school  purposes  ? 

4.  Taxes  were  levied  in  a  certain  town  for  the  following  pur- 
poses : — support  of  poor,  $2,000;  roads  and  bridges,  $500;  accounts 
audited  by  town  auditors,  $2,876.10  ;  accounts  audited  by  super- 
visors, $19.48;  county  expenses,  $9, 774. 72  less  a  surplus  of  $6,055.90 
in  the  county  treasury;  state  and  school  tax,  $15,079.88  ;  surplus 
tax,  $868.98.     What  was  the  rate  of  taxation,  the  total  valuation 
of  the  property,  as  made  by  the  town  assessors,  being  $4,321,252  ? 
What  was  the  tax  of  Mr.  A.,  whose  valuation  was  $7,300  ? 

NOTE. — To  save  labor  in  the  calculation  of  taxes,  a  table  similar  to  the 
following  is  usually  prepared  by  the  accountant. 

TAX  TABLE.— Rate,  5.8  mills  on  $1. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

.0580 

.0638 

.0696 

.0754 

.0812 

.0870 

.0928 

.0986 

.1044 

.1102 

2 

.1160 

.1218 

.1276 

.1334 

.1392 

.1450 

.1508 

.1566 

.1624 

.1682 

3 

.1740 

.1798 

.1856 

.1914 

.1972 

.2030 

.2088 

.2146 

.2204 

.2262 

4 

.2320 

.2378 

.2436 

.2494 

.2552 

.2610 

.2668 

.2726 

.2784 

.2842 

5 

.2900 

.2958 

.3016 

.3074 

.3132 

.3190 

.3248 

.3306 

.3364 

.3422 

6 

.3480 

.3538 

.3596 

.3654 

.3712 

.3770 

.3828 

.3886 

.3944 

.4002 

7 

.4060 

.4118 

.4176 

.4234 

.4292 

.4350 

.4408 

.4466 

.4524 

.4582 

8 

.4640 

.4698 

.4756 

.4814 

.4872 

.4930 

.4988 

.5046 

.5104 

.5162 

9 

.5220 

.5278 

.5336 

.5394 

.5452 

.5510 

.5568 

.5626 

.5684 

.5742 

5.  Find  from  the  above  table  the  tax  on  $16750. 

OPERATION.  ANALYSIS. — By  looking  in  the  table  oppo- 


Tax  on  $16000  is  $92.80 
750  "      4.35 


16750 


site  1  and  under  6,  we  find  that  the  tax  on 
$16  is  $.0928,  and  by  removing  the  point  3 
places  to  the  right,  we  find  the  tax  on 
$16000  to  be  $92.80.  In  the  same  manner, 


97.15 

the  tax  on  $750  is  found  to  be  $4.35.     The  tax  on  $16750  is  $92.80  plus 
$4.35,  or  $97.15. 


278  TAXES.  [Art.  611. 

6.  How  much  was  paid  by  Mr.  B.  on  an  assessment  of  $6400, 
the  collector  charging  a  commission  of  \%  additional  ?  (Use  table.) 

7.  Mr.  D.  being  delinquent  was  charged  5%  additional.     How 
much  was  he  obliged  to  pay  on  a  valuation  of  $9500  ? 

8.  What  was  the  total  tax,  including  commission  of  1%  of 
Mr.  C.,  whose  real  estate  was  assessed  at  $24000,  and  personal 
property  at  $15500  ? 

Find  the  tax  on  Find  the  tax  on 

9.  $7200  at  8.4  mills  on  $1.  18.  $248000  at  82.131  on  $100. 

10.  $27500  at  $2.174  on  $100.  14.  $14100  at  $16.476  on  $1000. 

11.  $4800  at  $21.871  on  $1000.  15.  $240500  at  $0.889  on  $100. 

12.  $9600  at  6.8  mills  on  81.  16.  $13500  at  $29.142  on  $1000. 

17.  A  has  6  lots  worth  $1200  each  :  B  has  8  lots  worth  $1500 
each;  and  C  has  10  lots  worth   $1000   each.     Divide   a  tax  of 
$1360  for  street  improvements  between  them. 

18.  A  rate  of  $6.34  on  $100  produces  a  tax  of  $4216.10.     What 
is  the  property  assessed  at  ? 

19.  A  pays  $17.25  more  tax  than  B,  their  valuations  being 
equal.     Living  in  different  towns,  the  rates  of  taxation  are  $3.176 
and  $3.291  on  $100  respectively.     What  is  the  valuation  of  their 
property  ? 

20.  The  cost  of  a  new  school  house  was  $3800.     What  was  the 
rate   of   the   tax  on  $100,    the   valuation  of  the    district   being 
$325000  ? 

21.  If  the  assessor's  valuation  of  certain  property  is  60$  of  its 
actual  value,  and  the  tax  is  2}^  of  the  assessment,  the  tax  is  what 
^  of  the  actual  value  ? 

22.  In  the  City  of  Brooklyn,  N.  Y.,  the  following  is  the  rule 
regarding  the  payment  of  taxes  : 

Rebate  at  the  rate  of  *l-fa%  per  annum  allowed  on  payments  made  during 
the  month  of  December  for  the  unexpired  portion  thereof.  On  all  bills  paid 
after  Jan.  1,  interest  at  the  rate  of  9%  per  annum  will  be  added,  to  be  com 
puted  from  Dec,  1  to  the  date  of  payment. 

According  to  the  above  rule,  how  much  tax  was  paid  Dec.  16, 
by  Mr.  A.,  the  valuation  of  whose  property  was  $7500,  the  rate  of 
tax  being  $2.376  per  $100  ?  How  much  was  paid  by  Mr.  B.,  on  a 
valuation  of  $12500,  Mar.  26  ?  (365  days  to  the  year.) 

23.  What  is  the  total  tax  on  8375  pounds  tobacco  at  8c.,  4360 
gallons  distilled  spirits  at  70c.,  2165  barrels  beer  at  $1  ? 


DUTIES. 


618.  Duties  or  Customs  are  taxes  assessed  by  the  Govern- 
ment upon  imported  merchandise  for  the  purpose  of  revenue  for 
the  support  of  the  government  and  for  the  protection  of  home 
industry. 

1.  The  waters  and  shores  of  the  United  States  are  divided  into  collection 
districts ;  in  each  of  which  there  is  a  port  of  entry  and  one  or  more  ports  of  de- 
livery.    Thus,  the  district  of  Boston  and  Charlestown  comprises  all  the  waters 
and  shores  within  the  counties  of  Middlesex,  Suffolk,  and  Norfolk.    Hingham, 
Weymouth,  Cambridge,  Roxbury,  and  Dorchester,  the  ports  of  delivery.    All 
ports  of  entry  are  also  ports  of  delivery. 

2.  The  principal  officer  of  every  district  is  the  collector,  who  is  assisted 
by  deputy-collectors,  surveyors,  appraisers,  weighers,  gangers,  inspectors,  etc. 
The  duties  of  the  above  vary  in  the  several  collection  districts  and  ports. 
There  is  also  in  the  leading  ports  of  entry,  a  "  naval  officer,"  whose  depart- 
ment is  a  check  upon  that  of  the  collector. 

619.  A  Custom-House  Broker  is  a  person  who  makes  entries, 
secures  permits,  and  transacts  other  business  at  custom-houses  for 
merchants.     He  is  familiar  with  the  tariif  laws  and  the  details 
and  regulations  of  custom-house  business,  and  usually  acts  under 
a  power  of  attorney. 

620.  Duties  are  of  two  kinds,  ad  valorem  and  specific. 

621.  An  Ad  valorem  Duty  is  a  tax  assessed  at  a  certain 
per  cent,  on  the  dutiable  value  of  the  merchandise;  as  silks  at  60%, 
watches  at  25%,  linens  30,  35  and  40%,  china  45  and  50%. 

1.  The  dutiable  value  of  merchandise  is  its  market  value  at  the  port  of 
export,  but  not  less  than  its  invoiced  cost,  commission  added,  whether  paid 
or  not.    There  is  no  duty  on  the  freight  or  transportation  from  the  port  of 
export.     The  appraised  value  is  sometimes  greater  than  the  invoice  value 
(623). 

2.  In  reducing  foreign  money  to  TJ.  S.  money  for  the  purpose  of  calcu- 
lating duties,  if  the  cents  of  the  result  are  less  than  50,  they  are  rejected  :  if 
more  than  50,  $1  is  added  to  the  dollars. 


280  DUTIES.  [Art.  622 

622.  A  Specific  Duty  is  a  tax  assessed  at  a  certain  sum  per 
ton,  pound,  foot,  yard,  gallon,  or  other  weight  or  measure,  with- 
out reference  to  the  value ;  as  leaf  tobacco  at  35^  per  pound,  clay 
$5  per  ton,  plate  glass  per  square  foot,  brandy  $2  per  proof  gallon, 
lumber  per  M  feet  board  measure,  salt  (in  bulk)  8  cts.  per  100 
Ibs.,  cotton  goods  per  square  yard. 

1.  Before  specific  duties  are  calculated,  allowances  are  made  for  tare  (the 
weight  of  the  box,  barrel,  or  cask),  leakage  (of  liquids  in  barrels),  and  breakage 
(of  liquids  in  bottles,  usually  5%). 

2.  The  U.  S.  Custom  House  ton  contains  2240  Ibs.  (336,  3)  the  hundred- 
weight 112  Ibs.,  and  the  quarter  28  Ibs. 

3.  On  certain  goods,  there  is  both  a  specific  and  an  ad  valorem  duty  (some- 
times called  a  combined  duty) ;  as  statuary  marble,  $1  per  cubic  foot  and  25  % , 
woollen  goods,  50  cts.  per  pound  and 


623.  An  Invoice  (418)  is  a  statement  made  by  the  seller  or 
shipper  of  merchandise  giving  a  description  of  the  same,  and  show- 
ing marks,  numbers,  quantity,  value,  charges,  and  other  details. 
(See  Ex.  24,  Art.  628.) 

1.  All  invoices  shall  be  made  out  in  the  weights  and  measures  of  the 
country  from  which  the  importation  is  made. 

2.  All  invoices  of  merchandise   subject  to  a  duty  ad  valorem,  shall   be 
made  out  in  the  currency  of  the  country  or  place  from  whence  the  importation 
is  made. 

3.  When  the  value  of  the  foreign  currency  is  fixed  by  law  (see  Art.  566), 
the  value  is  to  be  taken  in  estimating  the  duties;  when  the  value  is  not  fixed 
by  law,  the  invoice  must  be  accompanied  by  a  consular  certificate  showing  its 
value. 

624.  An  Appraiser  is  an  officer  of  the  customs  who  ex- 
amines imported  merchandise  and  determines  the  dutiable  value 
and  the  rate  of  duty  of  the  same. 

1.  The  place  where  the  examinations   are  usually  made  is  called  the 
"Public  Store." 

2.  One  package  of  every  invoice,  and  one  package  at  least  out  of  every 
ten  similar  packages,  shall    be   sent   to   the   public   store  for  examination. 
Certain  bulky  and  heavy  articles  are  examined  at  the  wharf  where  unloaded. 
Weighable  and  gaugeable  goods  on  which  the  duties  are  specific,  are  not  sent 
to  the  public  store  for  examination. 

3.  When  the  appraised   value   of  any  merchandise  subject   to  an  ad 
valorem  duty  is  10%  more  than  the  invoice  value  as  entered  by  the  importer, 
then  in  addition  to  the  duty  imposed  by  law  on  the  same,  there  shall  be  col- 
lected 20%  of  the  duty  imposed  on  the  same. 


Art.  625.]  D  UTIES.  281 

625.  A  Bonded  "Warehouse  is  a  place  for  the  storage  of 
merchandise  on  which  the  duties  or  taxes  have  not  been  paid. 

1.  If  an  importer  does  not  desire  to  place  his  goods  at  once  in  the  market, 
or  anticipates  exporting  the  same,  by  giving  a  bond  for  the  payment  of  the 
duties  and  making  the  entry  in  the  proper  form,  he  may  have  the  merchan- 
dise stored  at  his  own  risk  in  a  bonded  warehouse,  and  thus  defer  the  payment 
of  the  duties. 

2.  Merchandise  may  be  withdrawn  from  a  bonded  warehouse  for  expor- 
tation to  Canada  or  other  foreign  country,  without  the  payment  of  the  duty 
on  the  same. 

3.  Merchandise  is  frequently  sold  "in  bond"  at  prices  which  do  not  in- 
clude the  duty. 

4.  Merchandise  that  may  be  in  warehouse  under  bond  for  more  than  one 
year,  will  be  liable  when  withdrawn  for  10$  additional  duty. 

5.  Any  goods  remaining  in  public  store  or  bonded  warehouse  beyond 
three  years  shall  be  regarded  as  abandoned  to  the  government,  and  sold  under 
certain  regulations  and  the  proceeds  paid  into  the  Treasury. 

626.  Drawback,  —  When  distilled  spirits,  fermented  liquors 
and  tobacco  upon  which  an  internal  revenue  tax  has  been  paid, 
and  foreign  merchandise  upon  which  an  import  duty  has  been 
paid,  are  exported,  the  tax  or  duty  upon  the  same  is  refunded. 
Such  return  of  the  tax  or  duty  is  called  a  Drawback. 

627.  The  Free  List  is  a  list  of  articles  which  are  exempt 
from  duty. 

In  making  entries  of  free  goods,  the  value  as  given  in  foreign  money 
must  be  reduced  to  U.  S.  money  (See  Ex.  23,  Art.  628),  permits  must  bo 
obtained  to  land  the  goods,  and  certain  packages  are  sent  to  the  public  store 
for  examination. 

EXAMPLES. 

628.  1.  A  merchant  imported  from  Lyons  an  invoice  of  silk, 
the  dutiable  value  (621,  1)  of  which  was  48765  francs.     What 
was  the  dutiable  value  of  the  same  in  U.  S.  money,  and  what  was 
the  duty  at  60^  (621)  ? 

NOTES.  —  1.  For  foreign  moneys  of  account  and  their  values  in  United 
States  money,  see  Art.  566. 

2.  48765  francs  at  19.3^  =  $****.     (See  Art.  621,  2.)    60^  of  $****  = 


2.  Find  the  duty  on  1617  pounds   of  almonds,  at  6  cts.  per 
pound. 


282  DUTIES.  [Art.  628. 

3.  Find  the  total  duty  on  3  cases  machinery,  total  value  £23 
7s.  ±d.  at  45$ ;  7  cwt.  0  qr.  14  lb.  (622,  2)  castings  at  1J0.  per 
pound ;  4  cwt.  0  qr.  26  $.  chain  at  2Jc.  per  pound ;  and  3  cwt. 
1  qr.  4  Z#.  chain  at  2c.  per  pound. 

^.  An  invoice  of  woollen  cloth  weighing  516  pounds,  and 
valued  at  £327  165. ,  was  imported  from  England.  What  was  the 
duty  at  50  cts.  per  pound  and  35$  ? 

5.  An  importer  on  making   his  entry  at  the  custom-house, 
paid  the  duty  on  38716  pounds  (Invoice  weight)  of  tobacco,  at 
35  cts.  per  pound.     According  to  the  return  of  the  custom-house 
weigher,  the  net  weight  was  38472  pounds.     How  much  of  the 
duty  was  refunded  when  the  entry  was  liquidated  ? 

6.  The  duty  on  28432  pounds  of  sugar  was  paid  at  the  rate  of 
2f  cts.  per  pound.     According  to  the  weigher's  return,  the  net 
weight    was   28218   pounds.      How   much   additional   duty   was 
collected,  the  appraiser  having  fixed   the  duty  at   3J   cts.    per 
pound  ? 

7.  What  is  the  duty  on  an  invoice  of  linens  from  Ireland, 
dutiable  value  £424  15s.  6(7.,  at  35%  ?     Dutiable  value  £384  14s. 
§d.,  at  40$  ? 

8.  What  is  the  duty  on  an  invoice  of  porcelain  vases  from 
Paris  at  50$,  dutiable  value  9843  francs  ?    Dutiable  value  7896 
francs,  at  40$  ? 

9.  Find  the  duty  on  475  cu.  ft.  of  statuary  marble  imported 
from   Italy,    dutiable   value   16425   lire,    at   $1   per   cubic   foot, 
and  25$. 

10.  What  is  the  duty  on  37420  pounds  of  pig  iron  at  $7  per 
ton  (622,  2)  ? 

11.  Find  the  duty  on  an  invoice  of  leather  goods  from  Vienna, 
dutiable  value  6429  florins,  at  35$. 

12.  What  is  the  duty  on  an  importation  of  toys  from  Germany, 
dutiable  value  8437  marks,  at  50$  ?    Dutiable  value  7416  marks, 
at  45$  ? 

IS.  What  is  the  duty  at  28  cents  per  sq.  yd.  and  35$,  on 
1248  yards  of  Brussels  carpet,  27  in.  wide,  invoiced  at  3s.  6d.  per 
yard,  shipping  charges  (less  consul's  fee)  £2  16s.  9d.? 

14.  Find  the  duty  on  an  importation  from  Canada  of  5284 
bushels  of  potatoes,  invoiced  at  45  cts.  per  bushel,  and  37475 
pounds  of  hay,  invoiced  at  $12.50  per  ton  (2000  Ibs.),  the  duty  on 
potatoes  being  15  cts.  per  bushel,  and  on  hay 


Art.  628.] 


D  UTIE S. 


283 


15.  On  a  certain  invoice  of  34216  pounds  of  pepper,  there  are 
discounts  for  damage  as  follows :   12%  on  6190  pounds,  8%  011 
6438  pounds,  and  5%  on  9642  pounds.     After  deducting  the  dis- 
count, what  would  be  the  duty  on  the  remainder  at  5  cents  per 
pound  ? 

16.  The  duty  on  burlaps  is  30%  ad  valorem.      What  is  the 
amount   chargeable   on   a   bale  containing  50   webs,   each  being 
54  yds.  and  16  in*  long,  and  27  in.  wide,  and  valued  at  30  cents  per 
sq.  yd.? 

17.  What  is  the  amount  of  duty  chargeable  on  2465  pounds  of 
wool,  valued  at  £171  8s.,  when  the  rate  of  duty  is  10  cts.  per  pound 
and  11%  ad  valorem? 

18.  The  duty  on  certain  glass  plates  being  35  cents  per  sq.ft., 
find  the  duty  on  316  boxes,  each  containing  20  plates,  and  each 
plate  being  24  in.  by  30  in. 

19.  Find  the  duty  at  25%,  on  one  engraving,  cost  in  London 
£34  5s.,  case  and  shipping  charges  15s.,  commission  2J%. 

W.  What  is  the  duty  at  $1  per  cu.ft.  and  25%,  on  a  block  of 
marble  2x3x7  ft.,  imported  from  Italy,  dutiable  value  3450 
lire? 

21.  Find  the  duty  on  4175  Ibs.  cloves  at  5^  per  ».,  476  Us. 
cinnamon  at  20^,  and  5437  Ibs.  rice  at  2J-0. 

Make  the  extensions,  find  the  dutiable  value,  and  calculate  the 
duty  on  the  following  invoices  and  accompanying  entries : 

22.  Entry  of  merchandise,  imported  by  TEFFT,  WELLER  &  Co., 
from  Berlin  in  the  Str.  "  Silesia."  Arrived  Jan.  14,  1882.  New 
York,  Jan.  16,  1882. 


Marks. 


Nos. 


351 


Packages  and  Contents. 


One  case  half  silk  goods, 
Commission  2|%, 


Em.  ****.** 

60%  of  $*** 


=      $ 


*** 


Em.  2399.80 


284 


D  UTIES. 


[Art.  623. 


NOTE. — The  following  is  an  entry  of  free  goods.  Free  goods  are  entered 
and  the  foreign  monetary  units  reduced  to  U.  S.  money  for  statistical  purposes 
in  the  same  manner  as  dutiable  goods. 

23.  Entry  for  consumption  of  merchandise,  imported  by  W.  H. 
ScHiEFFELii*  &  Co.,  in  the  Str.  "  Ailsa"  from  Savanilla,  on  the 
10th  day  of  January,  1882.  New  York,  Jan.  12,  1882. 


Free. 


33  bales  Medicinal  Bark,      ......       2310. 

Packing, 12. 

Commission  2J$, **  ** 

(Pesos  of  TL  S.  of  Columbia), . .  ****  ** 

@  82.3^, 

24.  Invoice  of  one  package  merchandise,  purchased  by  GLAD- 
HILL  &  Co.  for  account  of  D.  BUCKLEY  &  Co.,  New  York,  and 
forwarded  for  shipment  to  D.  &  C.  MAC!VEK,  Liverpool. 

£.      *.      d. 

D.  B.    4  Pieces  Drab  Cotton  Pantaloon  32  in.  wide,  . 
207       31729     79yd, 

30  80, 

31  77$, 

32  79,   315 £  (less  fr)  307  @  2s.  2d.,  **     *     * 

discount, ** 

Verification  and  Commissioner's  fee,      .  14   10 

%\%  Commission,  . 16     5 

**     *     * 

Less  Consul's  Certificate  (not  dutiable),  14   10 

33    iT    7 

Entry  of  merchandise,  imported  by  D.  BUCKLEY  &  Co.  in  the 
Str.  "Catalonia"  from  Liverpool.  New  York,  Jan.  12,  1882. 

D.  B.         One  case  cotton, 33-11-7 

207  @  4.8665  "  **** 

Duty  35^  of  $***  =          I**.** 


Art.  628.] 


DUTIES. 


285 


25.  Invoice  of  700  bales  leaf  tobacco  shipped  by  F.  B.  DEL  Rio 
&  Co.,  per  Str.  "  Niagara "  for  New  York,  and  consigned  to 
FREDERICK  DE  BARY  &  Co. 


F.  B.          700  bales  83077  Ibs.  (See  page  343,  Spain) 

CHARGES. 
4027  Baling,        .......    $525. 

Export  duties,      .....    3407.39 

Consul  fee,      ......         2.75 

Small  charges,      .....       49 


Commission 
Spanish  gold 
HAVANA,  Dec.  27,  1881. 


$35000 


**** 


*** 


I***** 


** 
** 
** 


Custom  House,  New  York,  Collector's  Office,  Jan.  4,  1882. 
Bond  No.  9817. 

Entry  of  merchandise,  imported  on  the  third  day  of  January, 
1882,  by  FREDERICK  DE  BARY  &  Co.,  in  the  Str.  "Niagara" 
from  Havana. 


Marks. 

Nos. 

Packages  and  Contents. 

35c. 

F  B 

3828 

700  bales  Leaf  Tobacco 

84240  Ibs. 

$39958.74 

Duty  84240  Ibs.  @  35^                   =  $*****. 
f  Weighers  return  83675  Ibs.  at  35,^  =    *****.** 

@  .93,2= 

Refund,       ....         $***.** 
t  Added  by  the  liquidator. 

26.  What  is  the  duty  on  an  invoice  of  crockery  invoiced  at 
£1275  16s.  6d.  /.  o.  b.  (free  on  board),  at  40$  ? 

27.  What  is  the  duty  on  28916  pounds  of  steel  rails  at  1J^  per 
pound,  and  11438  pounds  of  tin  plates  at  1^  per  pound  ? 

28.  The  duty  on  spool  thread  of  cotton,  containing  100  yds.  to 
the  spool,  is  60  per  dozen  spools  and  in  addition  thereto  30^  ad 
valorem.     What  is  the  duty  on  11160  spools  valued  at  3^  a  spool? 


PARTNERSHIP. 


629.  Partnership  is  the  association  of  two  or  more  persons 
who  join  their  capital  and  services  for  the  purpose  of  conducting 
business,  the  gains  or  losses  being  shared  in  such  proportion  as 
may  be  stipulated  in  the  agreement. 

The  business  association  is  called  a  Firm,  House,  or  Company  ;  and  each 
individual  of  the  association  is  called  a  Partner. 

630.  A  Special  Partner  is  one  who  takes  no  active  part  in 
the  business,  and  whose  liability  is  limited  to  the  amount  of  his. 
investment.     In  order  to  thus  limit  his  liability,  the  amount  of 
his  investment  ^  must  be  duly  advertised,  and  he  must  take  no 
active  part  in  the  business. 

The  partners  who  conduct  the  business  are  called  General 
Partners.  Their  private  property  is  liable  for  the  debts  of  the 
partnership. 

631.  The  Capital  or  Capital  Stock  is  the  money  or  other 
property  which  is  invested  in  a  business. 

The  partners'  accounts  are  used  to  show  the  amounts  invested. 

In  most  firms,  the  investments  are  entered  in  the  partners'  "  stock  ac- 
counts," and  the  amounts  withdrawn  by  the  partners  during  the  year  and 
their  salaries  are  entered  in  their  "private  accounts." 

632.  A  Resource  or  Asset  is  any  kind  of  property  belong- 
ing to  the  concern  having  a  financial  value. 

633.  A  Liability  is  a  debt  owing  by  the  concern. 

634.  The  Net  Worth  of  a  concern  is  the  excess  of  its 
resources  over  its  outside  liabilities. 

635.  The  Net  Insolvency  of  a  concern  is  the  excess  of  its 
outside  liabilities  over  its  resources.     The  concern  being  unable  to 
pay  its  debts  in  full,  it  is  said  to  be  insolvent. 


Art.  636.]  PARTNERSHIP. 


387 


636.  Gains  or  Losses,  how  shared. — In  most  partner- 
ships, the  gains  or  the  losses  are   divided   according  to    certain 
fractions  or  percentages ;  the  inequalities  of  the  investments  are 
adjusted  by  allowing    interest   upon   the    same ;   and   the   part- 
ners receive  salaries  for  their  services  rendered.      (See  Ex.  36, 
Art.  639.)     Sometimes  the  net  gain  or  net   loss   is   shared   in 
proportion   to  the   investments    (Ex.    13,    Art.   639),    or    the 
average   investments.      (Ex.    17,   Art.   639.)      In    joint    stock 
companies   the   gains   (dividends)   and  the   losses    (assessments) 
are  shared  in  proportion  to  the  investments  or  the  amounts  of 
stock  held. 

637.  Gains  or  Losses,  how  found. — When  the  books 
have  been  kept  by  "  Single  entry/'  and  when  no  books  have  been 
kept,  the  gain  is  found  by  subtracting  the  net  worth  (634)  at 
commencing,  or  the  investment,  from  the  net  worth  at  closing ; 
and  the  loss,  vice  versa. 

When  the  books  have  been  kept  by  "  Double  entry,"  the  gain 
may  be  found  as  above,  or  by  subtracting  the  sum  of  the  separate 
losses  from  the  sum  of  the  separate  gains.  The  results  by  the 
two  methods  should  be  the  same  and  should  prove  each  other. 


EXERCISES. 

638.  In  the  following  exercises  find  the  gain  or  the  loss  : 

1.  Capital  at  commencing,  $5000  ;  capital  at  closing,  $3000. 

2.  Capital  at  commencing,  $5000  ;  capital  at  closing,  $8000. 


3 

$1000 


apa  a   commencng,  ;  capa  a   cosng,  . 

Capital  aj;    commencing,    $5000  ;    insolvency    at    closing, 
. 

4.  Capital  at    commencing,    $5000  ;    insolvency    at  closing, 
00. 

5.  Insolvency    at    commencing,    $5000  ;    capital  at   closing, 
00. 

6.  Insolvency  at  commencing,    $5000  ;    capital    at    closing, 
00. 

7.  Insolvency  at  commencing,  $5000  ;  insolvency  at  closing, 
$4000. 

8.  Insolvency  at  commencing,  $5000  ;   insolvency  at  closing, 
$9000. 


$7000 
5. 

$2000 
6. 

$6000 


288  PARTNERSHIP.  [Art.  638. 

Find  the  capital  or  the  insolvency  at  closing  : 

9.  Capital   at   commencing,  $5000  ;   gain   during   the  year, 
$3000. 

10.  Capital   at   commencing,  $5000  ;   gain   during  the  year, 
$6000. 

11.  Capital   at   commencing,  $5000  ;    loss   during   the   year, 
$2000. 

12.  Capital   at   commencing,  $5000  ;    loss   during  the   year, 
$8000. 

IS.  Insolvency  at  commencing,  $5000 ;  gain  during  the  year, 
$1000. 

14.  Insolvency  at  commencing,  $5000 ;  gain  during  the  year, 
$7000. 

15.  Insolvency  at  commencing,  $5000 ;    loss  during  the  year, 
$4000. 

16.  Insolvency  at  commencing,  $5000 ;    loss  during  the  year, 
$9000. 

Find  the  capital  or  the  insolvency  at  commencing  : 

"'x|7.  Capital  at  closing,  $5000  ;  gain  during  the  year,  $3000. 

18.  Capital  at  closing,  $5000  ;  gain  during  the  year,  $6000. 

19.  Capital  at  closing,  $5000  ;  loss  during  the  year,  $4000. 

20.  Capital  at  closing,  $5000  ;  loss  during  the  year,  $9000. 

21.  Capital  at  closing,  $8400  ;  gain  during  the  year,  $4100. 

22.  Capital  at  closing,  $3700 ;  gain  during  the  year,  $5200. 
28.  Insolvency  at  closing,  $5000 ;  gain  during  the  year,  $1000. 
&£.  Insolvency  at  closing,  $5000  ;  gain  during  the  year,  $8000. 

25.  Insolvency  at  closing,  $5000  ;  loss  during  the  year,  $2000. 

26.  Insolvency  at  closing,  $5000  ;  loss  during  the  year,  $7000. 

EXAMPLES. 

639.  1.  A  and  B  are  partners,  A  sharing  f  of  the  gain  or  loss 
and  B  \.  A  invests  $5000,  and  B  $2350.  At  the  end  of  the  year 
their  resources  and  liabilities  are  as  follows  :  merchandise  on  hand, 
per  inventory,  $2000 ;  real  estate,  $7000 ;  cash  on  hand  and  in 
bank,  $1532  ;  due  on  personal  accounts,  $1640.25  ;  notes  on  hand, 
$1000  ;  notes  outstanding,  $800  ;  owing  by  the  concern  to  sundry- 
persons,  $4471.69.  What  is  the  amount  of  net  resources  belonging 
to  each  partner  ? 


Art.  639  ] 


PARTNERSHIP. 


289 


FIRST  OPERATION. 


$2000 
7000 
1532 
1640.25 
1000    $13172.25 


RESOURCES. 

Merchandise  on  hand,  . 
Real  estate,  .... 
Cash  on  hand, 
Personal  accounts, 
Bills  receivable,    . 

LIABILITIES. 
Bills  payable, 
Personal  accounts, 
Present  worth, 
Investments  (subtracted), 

Total  net  gain, 

f  of  $550.56  =  $367.04,  A's  share  of  the  gain. 
£  of  $550.56  =    183.52,  B's  share  of  the  gain. 


$800 
4471.69 


5271.69 
$7900.56 
7350. 
$550.56 


A's  investment, 
Plus  his  gain, 

A's  present  worth, 


$5000 
367.04 


B's  investment, 
Plus  his  gain, 

B's  present  worth, 


$2350. 

183.52 


$5367.04          B's  present  worth,       $2533.52 
$5367.04  +  $2533.52  =  $7900.56,  total  present  worth,  as  above. 

SECOND  OPERATION. 

ANALYSIS. — Theoretically,  all  the  resources  of  a  business  belong  to  the 
creditors  and  the  partners  (proprietors),  the  partners'  investments  being 
regarded  as  liabilities  ;  hence,  the  resources  and  liabilities — including  the 
partners'  accounts — should  be  equal.  If,  in  a  statement  of  the  condition  of  a 
business,  the  resources  and  liabilities  thus  considered  should  not  be  equal,  it 
is  evident  that  the  partners'  accounts  do  not  show  their  true  interests,  and  the 
inference  is  that  a  gain  or  loss  has  occurred  which  has  not  been  entered  to 
their  accounts.  The  excess  of  resources  over  liabilities  would  in  such  case 
show  the  gain,  as  would  the  excess  of  liabilities  over  resources  show  the  loss. 
In  order  to  restore  the  equilibrium,  the  gain  should  be  credited  or  the  loss 
debited  to  the  partners'  accounts. 

1.  STATEMENT  BEFORE  ADJUSTING  PARTNERS'  ACCOUNTS. 


RESOURCES. 
Merchandise, 
Real  estate, 
Cash, 

Personal  accounts,     . 
Bills  receivable, 


2000 

7000 

1532 

1640.25 
_1000_ 
13172.25 
12621.69 


Excess  of  resources  (net  gain).     550.56 


LIABILITIES. 

Bills  payable, 
Personal  accounts, 
A's  investment, 
B's        do. 


800 

4471.69 
5000 
2350 

12621.69 


A's  |,  $367.04  ;  B's  fc  $183.52. 


290 


PAR  TNE R  SHIP. 


[Art.  639. 


2.  STATEMENT  AFTER  ADJUSTING  PARTNERS'  ACCOUNTS. 


RESOURCES. 
Merchandise,     . 
Real  estate, 
Cash, 

Personal  accounts,    . 
Bills  receivable, 


2000 

7000 

1532 

1640.25 

1000 


13172.25 


LIABILITIES. 

Bills  payable,    . 
Personal  accounts,    . 
A's  investment  and  gain, 
B's          do.  do. 


800 

4471.69 
5367.04 
2533.52 

13172.25 


2.  A  and  B  are  partners,  A  sharing  f  of  the  gain  or  loss  and 
B  J.  A  invested  $5000,  and  B  $2350.  During  the  year  the  con- 
cern gained  on  merchandise,  $955.56  ;  on  real  estate,  $315.  The 
expense  account  showed  a  loss  of  $675  ;  the  interest  account,  $45. 
What  was  the  net  gain,  and  the  balance  of  each  partner's 
account. 

NOTE. — The  above  example  is  the  complement  of  Ex.  1.  The  books 
having  been  kept  by  double  entry,  the  separate  gains  and  losses  are  given, 
and  the  net  gain  thus  found.  The  loss  and  gain  account  and  the  partners' 
accounts  are  shown  in  the  following  operation  in  "  skeleton  ledger  "  form. 

OPERATION. 
A.  B. 


Balance, 

5367 

04 

Investment, 
Gain,    .    . 

5000 
367 

04 

5367 

04 

5367 

04 

Balance,  .    i  5367 

04 

Balance, 

2533 

52 

Investment, 
Gain,    .    . 

2350 
183 

2533 

52 
52 

2sas 

52 

Balance,  . 

2533 

52 

Loss  AND  GAIN. 


Expense,    .    . 

675 

Mdse.,      .    . 

955 

56 

Interest,     .    . 

45 

Real  Estate, 

315 

A's,Gainl,    . 

367 

04 

B's     "     i,    . 

183 

52 

1270 

56 

1270 

56 

3.  A  and  B  started  in  business  July  1,  1881.  Each  put  into 
the  concern  $2200.  The  resources  on  Jan.  1,  1882,  were  as  fol- 
lows :  goods,  $4000  ;  bills  receivable,  $1500.  The  liabilities  were 
$580.  A  has  drawn  out  cash,  $3000  ;  and  B,  $2000.  How  much 
is  due  each  partner,  the  gain  or  loss  being  divided  equally  ? 

NOTES.- — 1.  It  must  be  borne  in  mind  that  the  amounts  drav/n  out  by  the 
partners  are  as  fully  resources  of  the  business  as  if  charged  to  an  outside 
party. 

2.  In  examples  3,  4,  5,  and  7,  open  accounts  with  the  partners  and  make 
statements  of  resources  and  liabilities. 


Art.  639.]  PARTNERSHIP.  291 

4-  A  and  B  are  partners.  They  have  cash  and  notes  on  hand 
to  the  amount  of  $6475.28.  A  has  drawn  from  the  concern  $2478.30, 
and  B  has  drawn  $1016.48.  A  invested  $4287.46,  and  B,  $1037.75. 
The  firm  owes  sundry  persons  $5016.82.  What  is  each  partner's 
present  interest  in  the  concern,  if  they  share  equally  in  gains  and 
losses  ? 

5.  On  Jan.  1,  my  brother  and  I  started  a  business  in  which  I 
invested  $900,  and  he  $400.     We  now  propose  to  separate,  and  the 
business  stands  as  follows  :  stock  in  store  $1800  ;  cash  on  hand 
and   in    bank,  $1200 ;    outstanding    accounts,  considered   good, 
$1200.     According  to  the  agreement,  I  am  entitled  to  f  of  the 
net  gain,  and  my  brother  J.     During  the  time  of  the  copartner- 
ship, I  have  drawn  $4000  and  he,  $2800.     Of  the  assets  given 
above,  how  much  are  we  each  entitled  to  ? 

6.  C,  D,  and  E  are  partners,  each  investing  $10000,  and  each 
to  share  |  of  the  gain  or  loss.     The  resources  and  liabilities  at 
the  close  of  business  are  found  to  be  as  follows,  viz. :  Merchandise 
on  hand,  per  inventory,  $8159.50  ;  cash  on  hand,  $5012.88 ;  per- 
sonal accounts  due  the  firm,  $4235 ;   notes  and  accepted  drafts 
(bills  receivable)  on  hand,  $5000  ;  real  estate,  $8000  ;  bonds  and 
stocks,   $12000  ;    owing  by  the  firm   to  sundry  persons,  $5505  ; 
firm's  notes  outstanding  (bills  payable),  $3000.     0  has  withdrawn 
during  the  year  $1247.87;    D,  $1400;   and  E,  $1489.     What  is 
each  partner's  interest  in  the  concern  at  closing  ? 

7.  C,  D,  and   E  are  partners,   sharing  the   gains  and  losses 
equally.     C's  net  investment  was  $8752.13  ;  D's,  $8600  ;  and  E's 
$8511.      During  the  year  the  firm's  gains  were  as  follows  :  Mer- 
chandise, $8529  ;  stocks  and  bonds,  $650  ;  interest,  $985.25.     The 
cost  of  conducting  the  business  was  $2125.     What  was  each  part- 
ner's interest  at  closing  ?     (See  Ex.  2.) 

8.  M  and  N  are  partners,  M  sharing  f  of  the  gain  or  loss  and 
N  £.     M  invested  $15000  and  N  $5000.     At  the  close  of  the  busi- 
ness year,  the  resources  and  liabilities  of  the  concern  are  as  fol- 
lows :  cash  on  hand,  $2128  ;  bills  payable,  $4000 ;  bills  receivable, 
$3000  ;  the  firm  owes  sundry  persons,  $8375;  due  the  firm  from 
sundry  persons,  $16427 ;  rent  paid  in  advance,  $375 ;  mortgage 
held  by  the  concern  on  the  property  of  A.  Gr.  Pope,  $5000;  accrued 
interest  on  the  same,  $150 ;  store  fixtures  valued  at  $835 ;  mer- 
chandise on  hand,  $9416 ;  accrued  interest  on  firm's  notes  out- 
standing, $112 ;  accrued  interest  on  notes  held  by  the  firm,  $75. 


292  PARTNERSHIP.  [Art.  639. 

M  lias  withdrawn  $2465;  and  N,  $2275.  According  to  the  agree- 
ment, each  partner  is  to  receive  a  salary  of  $2500.  What  are  the 
separate  interests  at  the  close  of  the  business  ? 

9.  A  owns  a  business,  the  good  will  of  which  is  estimated  at 
$10000,  and  the  stock  on  hand  at  $15000.     B  and  0  agree  tc 
unite  with  him  on  the  following  conditions  :   B  to  invest  $25000 
cash,  and  C  to  devote  his  entire  time  to  the  business,  for  which 
he  is  to  receive,  in  addition  to  his  interest,  an  annual  salary  of 
$1000.     The  capital  is  to  be  kept  intact,  and  no  interest  to  be 
allowed  therefor.     The  gain  or  loss  to  be  divided  equally  between 
the  three  partners.     At  the  end  of  the  year  the  resources,  includ- 
ing good  will,  book  accounts,  notes,  inventories,  etc.,  but  not  in- 
cluding amounts  drawn  by  the  partners,  amount  to  $67000,  and 
the  liabilities  to  outside  parties,  to  $10500.     C  has  drawn  out 
during  the  year  $2500 ;  B,  $1575  ;  A,  $2000.     Of  the  resources 
above  named  there  are  bad  debts  not  to  be  counted,  amounting  to 
$575.     What  is  the  condition  of  each  partner's  account  ? 

10.  A  and  B  are  partners,  A  investing  f  of  the  capital,  and 
B  J ;  the  gains  or  losses  to  be  shared  in  the  same  proportion. 
The  following  is  an  exhibit  of  the  business,  excepting  the  part- 
ners' accounts,  at  the  close  of  a  certain  period  :   Resources,  cash, 
$3775;  Stone  &  Co.,  $150;  A.  R.  Mead,  $1200;  bills  receivable, 
$5500  ;  interest  on  the  same,  $125  ;  merchandise,  $5140.     Liabil- 
ities, L.  Blair,  $500  ;  W.   H.   Rice,   $723  ;  Martens  &  Bultman, 
$517.64  ;  bills  payable,  $3300  ;  interest  on  the  same,  $169.     The 
net  gain  during  the  year  was  $3174.     What  was  each  partner's 
original  investment  ? 

11.  Upon  a  close  valuation  of  the  personal  accounts  due  the 
firm  in  the  preceding  example,  the  partners  are  convinced  that 
Stone  &  Co/s  is  worth  no   more  than   50%  of  its  face ;   and 
A.  R.  Mead's,  25%  of  its  face.     Upon  this  valuation  what  would 
be  the  gain,  and  what  the  condition  of  the  partners'  accounts  at 
closing  ? 

12.  A,  B,  and  C  are  partners,  A  investing  $25000  capital,  B 
$5000,  and  C  nothing.     The  proportionate  interests  are  :   A  60%, 
B  25%,  C  15%.     At  the  expiration  of  the  term  of  copartnership, 
and  after  the  gains  and  losses  have  been  adjusted,  A's  credit  of 
capital  stands  intact,  B  has  a  credit  of  only  $1000,  while  C  has 
overdrawn  his  account  $8534.     C  being  insolvent,  how  much  must 
B  pay  into  the  concern  to  adjust  his  share  of  the  loss  ? 


AH.  639.]  PARTNERSHIP.  293 

13.  A  and  B  are  partners  in  business,  the  gain  or  loss  to  be 

divided  in  proportion  to  investment.     A  invested  $8750  ;  B  in- 

vested $4000.     The  net  gain  is  $2726.15.  What  is  each  partner's 
share  ? 

FIRST  OPERATION.  —  FRACTIONAL  METHOD. 


ANALYSIS.—  Since  A's  investment,  $8750,  is  -fi$fo  of  tne  total  investment, 
he  is  entitled  to  $$fo  of  the  gain  ;  and  for  a  similar  reason,  B  is  entitled  to 
of  the  Sam- 

=  |f  ;  ff  of  $2726.15  =  $1870.89,  A's  gain. 
=  *  «  ;  *  «  of  $2726.15  =    $855.26,  B's  gain. 


SECOND  OPERATION.  —  BY  PROPORTION. 

ANALYSIS.  —  The  total  investment  is  to  each  partner's  investment  as  the 
total  gain  is  to  each  partner's  gain. 

$12750  :  $8750  :  :  $2726.15  :  $1870.89,  A's  gain. 
$12750  :  $4000  :  :  $2726.15  :    $855.26,  B's  gain. 

NOTE.  —  Cancel  any  factor  common  to  the  given  extreme  and  either  of 
the  means. 

THIRD  OPERATION.  —  BY  PERCENTAGE. 

ANALYSIS.—  $2226.15,  the  gain,  is  21.3816$  of  $12750,  the  total  invest- 
ment. The  partners'  gains  are  therefore  21.3816$  of  their  respective 
investments. 

21.3816$  of  $8750  =  $1870.89,  A's  gain. 

21.3816$  of  $4000  =    $855.26,  B's  gain. 

NOTE.  —  In  order  to  produce  exact  results  by  this  method,  it  is  necessary 
to  extend  the  number  expressing  the  rate  per  cent,  of  the  gain  or  loss  to 
several  decimal  places. 

14*  E,  F,  Gr,  and  H  enter  into  a  joint  speculation.  E  advances 
$5000,  F  $7000,  G  $8000,  and  H  $10000,  the  gain  or  loss  to  be 
divided  according  to  investment.  They  gain  $14285.  What  is 
the  share  of  each  ? 

15.  Four  merchants  ship  goods  on  joint  account.     A  puts  in 
$6000,  B  $5500,  C  $4200,.  and  D  $4800.     What  will  be  each  man's 
share,  if  the  gain  is  $9200  ? 

16.  Five  persons  having  claims  against  the  government,  placed 
their  claims  in  the  hands  of  an  agent  for  collection.     A's  claim 
amounted  to  $500,  B's  to  $425,  C's  to  $300,  D's  to  $250,  and  E's 
to  $175;  but,  after  the  agent  had  deducted  his  fees,   there  re- 
mained only  $1237.50.     How  much  did  each  claimant  receive  ? 


294 


PAR  TNE  R  SHIP. 


[Art.  639. 


17.  A  and  B  are  partners,  gain  or  loss  to  be  divided  in  pro- 
portion to  average  investment.  A  invests,  Jan.  1,  $4000 ; 
Mar.  1,  $2000  ;  Oct.  1,  $3000 ;  and  withdraws  July  1,  $1500  ; 
Dec.  1,  $1000.  B  invests,  Jan.  1,  $6000  ;  Sept.  1,  $3000.  They 
close  their  books  Jan.  1  of  the  following  year  and  find  they  have 
gained  $3456.  What  is  each  partner's  share  ? 

NOTE. — An  Average  Investment  is  an  investment  for  a  certain  period  of 
time  equivalent  to  several  investments  for  different  periods  of  time. 


OPERATION. 

A  invested  Jan.  1,  $4000  x  12  = 

Mar.  1,  2000  x  10  = 

Oct.  1,  3000  x  3  = 

A  withdrew  July  1,  1500  x  6  = 

Dec.  1,  1000  x  1  = 

A's  average  investment  for  1  month, 

OB, 


A  invested 


Jan.  1, 
Mar.  1, 


withdrew  July  1 
invested  Oct.  1, 
withdrew  Dec.  1, 


$4000 
2000 

6000 
1500 
4500 
3000 

7500 
1000 

6500 


2  = 


4  = 


3  = 


2  = 


x  i  = 


A's  average  investment  for  1  month, 


$48000 
20000 

9000   77000 
9000 

_1000   10000 
67000 


$8000 
24000 
11500 
15000 

6500 
$67000 


ANALYSIS. — By  the  first  operation,  we  suppose  each  investment  to  be 
made  for  the  remainder  of  the  time.  To  find  the  average  investment,  multi- 
ply each  investment  and  withdrawal  by  the  interval  between  its  date  and 
time  of  settlement.  Subtract  the  products  obtained  from  the  withdrawals 
from  the  products  obtained  from  the  investments.  The  remainder  will  be  the 
average  investment  for  1  month,  if  the  time  is  found  in  months.  A's  invest- 
ment of  Jan.  1  is  in  the  business  12  months  (Jan.  1  to  Jan.  1);  the  use  of 
$4000  for  12  months  is  equivalent  to  the  use  of  $48000  for  1  month.  Treating 
the  other  investments  in  like  manner,  we  find  A's  total  investments  are 
equivalent  to  $77000  for  1  month.  A's  withdrawals  are  equivalent  to  $10000 
for  1  month.  A's  net  average  investment  is  therefore  equivalent  to  $67000 
for  1  month. 

By  the  second  operation,  we  find  the  actual  amount  in  the  business  for 
each  month  of  the  year.  Jan.  1,  A  invested  $4000,  which  was  in  the  business 
until  Mar.  1,  or  for  2  months.  Mar.  1,  he  added  $2000,  making  his  total  invest- 


Art.  639.]  PARTNERSHIP.  295 

ment  $6000,  which  was  in  the  business  until  July  1,  or  for  4  months.  July  1, 
he  withdrew  $1500,  leaving  in  the  business  $4500  until  Oct.  1,  or  3  months,  etc. 
The  several  net  investments  as  found  in  this  manner  are  equivalent  to  $67000 
for  1  month. 

B's  average  investment,  as  found  by  either  of  the  above  methods,  is 
$84000  for  1  month. 

A's  average  investment  for  the  year  is  $5583.33^  ;  and  B's  $7000.  To 
avoid  fractions,  divide  the  gain  in  proportion  to  the  average  investments  for 
1  month.  After  the  average  investments  are  found  for  a  common  time,  the 
gain  may  be  divided  according  to  either  of  the  methods  under  Ex.  15.  By 
the  fractional  method,  A  would  be  entitled  to  T6/T  of  the  gain,  and  B  to  T8/T. 

18.  C  and  D  are  partners,  gain  or  loss  to  be  divided  in  propor- 
tion to  average  investment.     C  puts  in  $6000  for  one  year,  and 
$7000  for  one  and  a  half  years  ;  D  puts  in  $6000  for  two  and  a 
half  years.     The  net  loss  is  $1565.40.     What  is  each  one's  share  ? 

19.  A,  B,  and  C  are  partners.     A  puts  into  the  concern  $3000, 
but  withdraws  half  of  it  at  the  end  of  6  months  ;  B  puts  in  $2000, 
and  adds  $500  to  it  at  the  end  of  4  months  ;  C  puts  in  $2500  for 
the  whole  year.     The  gain  during  the  year  is  $1700.     What  is 
each  one's  share  ? 

20.  Three  contractors  agree  to  build  a  road  for  $10000.    A  has 
25  men  at  work  for  16  days  and  30  men  for  34  days.     B  has  40 
men  for  10  days  and  45  men  for  40  days.     0  has  48  men  for  50 
days.     C  receives  $200  extra  for  superintending  the  work.     How 
much  is  each  contractor  entitled  to  ? 

21.  J,  K,  and  L  are  partners,  gain  or  loss  to  be  divided  accord- 
ing to  average  investment.     J  invests  as  follows  :   Jan.  1,  $6000  ; 
Apr.   1,  $4000 ;   K  invests,  Jan.   1,  $8000 ;   L  invests,    Jan.    1, 
$7000  ;  Apr.  16,  $2500  ;  and  draws  out  June  16,  $3500.     At  the 
end  of  the  year  the  net  gain  is  found  to  be  $4135.60.     What  is 
each  partner's  share  ?     (Time  by  Compound  Subtraction.) 

22.  A,  B,  C,  and  D  were  partners  for  two  years.     When  the 
firm  commenced  business,  A's  investment  was  $6000,  B's  $3500, 
C's  $2800,  and  D's  $1700.     At  the  end  of  8  months,  A  withdrew 
$3000.     At  the  end  of  10  months,  D  added  $1300  to  his  former 
investment.     At  the  end  of  one  year,  B  withdrew  $800.     At  the 
close  of  the  two  years,  they  had  gained  $4727..    What  was  each 
partner's  share  of  the  gain  ? 

28.  A  and  B  are  partners  for  one  year,  the  gain  or  loss  being 
divided  in  proportion  to  their  average  investments.  A  invested, 
Jan.  1,  $8000 ;  June  16,  $1500  ;  Aug.  1,  $2500  ;  and  drew  out 


290  PARTNERSHIP.  [Art.  639. 

May  1,  $1500.  B  invested  Jan.  1,  $10000 ;  Apr.  1,  $500  ;  and 
withdrew  Aug.  16,  $2500.  How  much  should  A  invest  Sept.  1  to 
entitle  him  to  one-half  the  gain  ? 

24.  C  and  D  are  partners.  According  to  agreement  0  is  to 
share  f  of  the  gain  or  loss,  and  D  £.  At  the  end  of  the  year,  D 
desires  to  increase  his  investment  so  that  he  will  be  entitled  to  a 
J  interest.  How  much  must  D  invest,  the  partners'  accounts  after 
the  books  are  closed  being  as  follows:  C's  debit,  $6712.38;  C's 
credit,  $27000  ;  D's  credit,  $9000  ? 

25*  R,  S,  T,  and  U  enter  into  copartnership  with  equal  capital, 
upon  the  following  conditions  :  R  to  receive  as  a  salary  $2000 ; 
S,  $1500;  T,  $1200;  and  U,  $1000; 'the  gain  or  loss  to  be 
divided  equally.  At  the  close  of  the  year,  the  net  gain,  exclusive 
of  salaries,  proves  to  be  $5400.  To  how  much  of  this  amount  is 
each  entitled  ? 

NOTE. — The  excess  of  the  salaries  (losses)  over  the  gain  is  the  net  loss  to 
the  business,  and  should  be  charged  equally  to  the  partners. 

26.  X,  Y,  and  Z  commence  business  without  capital.     Accord- 
ing to  the  partnership  contract,  X  is  to  receive  a  salary  of  $3000  ; 
Y,  $2500 ;  and  Z,  $2000 ;  the  gain  or  loss  to  be  divided  equally. 
During  the  year,  X  withdraws  $3000 ;  Y,  $2800 ;  and  Z,  $1800. 
What  is  the  balance  due  each  partner,  if  the  gain,  without  taking 
into  account  the  partners'  salaries,  is  $9000  ? 

27.  A  merchant's  assets  are  $12000,  and  he  owes  A  $1900,  B 
$5000,  C  $3000,  and  D  $6100.     As  he  is  unable  to  pay  his  debts 
in  full,  he  is  compelled  to  make  an  assignment  for  the  benefit  of 
his  creditors.     If  the  expense  of  making  the  settlement  is  $1600, 
what  %  of  his  indebtedness  can  he  pay  ?    What  amount  will  each 
creditor  receive  ? 

28.  A  and  B  failed  in  business.     Their  liabilities  were  $64000. 
The   firm's   assets   amounted   to  $37500.     If   the  assignee's  fees 
and  other  expenses  were  $2300,  what  %  of  their  indebtedness  can 
they  pay  ?     What  will  C  receive,  whose  claim  is  $16400  ? 

29.  D  and  E  have  made  an  assignment.     They  owe  F  $4200, 
G  $16000,  H  $2500,  and   K  $11800.      Their  assets  amount   to 
$25400,  and  the  expense  of  making  the  settlement  was  $1630.     Gr 
being  a   preferred  creditor,    what   %   will  be  paid  to  the  other 
creditors,  and  what  amount  will  each  receive  ? 

NOTE. — A  preferred  creditor  is  one  who  is  paid  in  full,  before  any  divi- 
dend is  paid  to  the  other  creditors. 


Art.  639.]  PARTNERSHIP.  297 

SO.  A  lot,  whose  front  is  240  feet,  and  whose  depth  is  100  feet, 
is  bought  by  A,  B  and  C,  who  pay  respectively  $3000,  $4000,  and 
$5000.  How  many  feet  front  is  each  entitled  to,  if  it  is  divided  in 
proportion  to  their  investments  ? 

81.  M,  the  owner  of  a  mill,  employs  S,  a  miller,  under  the  fol- 
lowing conditions  :  M  is  to  furnish  the  requisite  capital,  and  S  to 
receive,  in  lieu  of  salary,  ^  of  the  profits.     M  has  a  store  connected 
with  the  mill,  on  the  books  of  which  are  entered  all  time  sales  of 
mill  products.     The  grain,  etc.,  for  the  mill  is  furnished  by  M. 
At  the  beginning  of  the  year  the  value  of  the  grain,  flour,  feed, 
etc.  is  $1727.     During  the  year  M's  purchases  for  the  mill  amount 
to  $19275.     S  has  received  for  cash  sales  $16337,  of  which  he  has 
paid  over  to  M  $15550.     The  sales  on  account,  as  shown  on  M's 
books,  amount  to  $8375;  and  the  value  of  the  products  on  hand  is 
$2828.     During  the  year  S  has  purchased  goods  at  M's  store  to  the 
amount  of  $837.65.     How  much  is  owing  to  S  at  the  expiration 
of  the  year  ? 

82.  P  and  Q  are  partners,  each  to  receive  interest  on  his  net 
investment  at  the  rate  of  §%  per  annum,  and  the  net  gain  or  loss 
to  be  divided  equally.     P  invests,  Jan.  1,  $5000  ;  Mar.  1,  $4000 ; 
June  16,  $1500;  and  draws  out  Apr.  16,  $2500.    Q  invests,  Jan.  1, 
$8000 ;  Sept.  16,  2000 ;  and  draws  out  June  1,  $1500 ;  Nov.  11, 
$500.     At  the  close  of  the  year  the  net   gain   is   found   to   be 
$4475.25,  without  taking  into  account  the  interest  on  the  part- 
ners' accounts.     What  is  the  amount  due  each  partner  after  the 
gain  is  adjusted  ?     (Time  by  Compound  Subtraction.) 

NOTE. — Open  accounts  with  P,  Q,  and  Loss  and  Gain. 

83.  A  and  B  have  been  doing  business  as  partners,  A  sharing 
|  and  B  f  of  the  gains  and  losses.     A  invested  $4500,  average  date 
Mar.  25,  1882  ;  and  drew  out  $2700,  average  date  Sept.  12,  1882. 
B  invested  $7200,   average  date  June  17,   1882  ;  and  drew  out 
$3750,  average  date  Oct.  25,  1882.     At  the  time  of  their  dissolu- 
tion, Jan.  1, 1883,  the  debts  of  the  firm  were  all  paid  and  they  had 
on  hand  belonging  to  the  firm  $8750  in  cash.      How  shall  the 
money  be   divided,    each  being   allowed   interest   at   6%  on   his 
investment  and  charged  with  interest  at  the  same  rate  on  the 
amounts  drawn  ?     (Exact  days.     Interest  360  days  to  the  year.) 

NOTE. — In  Examples  33  and  34,  open  accounts  with  A  and  B,  and  make 
statements  of  Resources  and  Liabilities. 


298  PARTNERSHIP.  [An.  630. 

34.  A  and  B  are  partners,  A  having  f  and  B  f  interest.  A 
advanced  in  business  $12000,  average  date  Jan.  12,  1883  ;  and 
drew  out  $1265,  average  date  Oct.  20,  1883.  B  advanced  $7500, 
average  date  Apr.  5,  1883 ;  and  drew  out  $2560,  average  date 
Nov.  25,  1883.  Jan.  1,  1884,  the  assets  are  as  follows  :  Cash, 
$5800 ;  merchandise,  $6250 ;  notes  on  hand,  $7300 ;  accrued 
interest  on  the  same,  $387.14;  personal  accounts,  $5700.  The 
liabilities  are  as  follows  :  Notes  outstanding,  $4200 ;  accrued 
interest  on  the  same,  $227.65 ;  personal  accounts,  $2500.  Find 
the  balance  of  each  partner's  account,  5%  of  the  personal  accounts 
being  considered  uncollectible,  and  interest  being  reckoned  on 
the  partners'  accounts  at  Q%  per  annum  (365  days  to  the  year). 

85.  A,  B,  and  0  form  a  copartnership  under  the  following  con- 
ditions :  A  is  to  manage  the  business,  and  to  receive  therefor  $2400 
per  annum,  which  amount  is  to  be  credited  as  July  1.     He  is  to 
receive  interest  on  his  salary  and  to  pay  interest  on  sums  with- 
drawn at  the  rate  of  6%  per  annum.     B  and  C  are  to  furnish  the 
capital,  and  to  receive  interest  therefor  at   the   rate  of   6%  per 
annum.     The  net  gain  or  loss  to  be  divided  equally.     B  invests, 
Jan.  1,  $10000  ;  Apr.  1,  $5000.     C  invests,  Jan.  1,  $10000 ;  July 
1,  $5000 ;  and  draws  out  Sept.  16,  $500.     A  draws  out,  Feb.  1, 
$200 ;  Mar.  1,  $400  ;  July  11,  $500 ;  Oct.  1,  $200;  Nov.  21,  $100. 
At  the  end  of  the  year,  the  gain — without  taking  into  account 
either  the  salary  to  be  paid  to  A  or  the  interest  on  the  partners' 
accounts — is  $8437.16.    What  will  be  the  balance  of  each  partner's 
account,  when  all  the  items  have  been  properly  entered  ? 

NOTE. — In  Examples  35  and  36  open  accounts  with  the  partners  and  with 
Loss  and  Gain. 

86.  A,  B,  and  0  are  partners,  A  sharing  -f  of  the  gain  or  loss, 
B  f ,  and  C  -^.     Interest  is  to  be  reckoned  at  the  rate  of  6%  per 
annum  (365  days  to  the  jear)  on  the  partners'  accounts,  and  each 
partner  is  to  receive  a  salary  of  $1800,  to  be  credited  as  July  1. 
A  invested,  Jan.  1,  $16000 ;  and  withdrew  during  the  year  $4875, 
average  date,  Aug.  21.     B  invested,  Jan.  1,  $20000  ;  and  withdrew 
$6224,  average  date,  June  18.     C  invested,  Jan.  1,  $5000 ;  and 
withdrew  $2625,  average  date,  July  31.     Jan.  1,  of  the  following 
year,  the  merchandise  account  shows  a  gain  of  $18437.16;  the 

( interest   account    (not   including  the   interest   on   the  partners' 
accounts)  a  gain  of  $586.38;  sundry  consignment  accounts  show 


Art.  639.]  PAR  TNER  S  HIP.  299 

a  net  gain  of  $1287.14.  The  expense  account  (not  including  the 
partners'  salaries)  shows  a  loss  of  $3424. 75.  What  is  each  partner's 
interest  in  the  business  at  closing  ?  How  will  A  be  affected  if 
each  partner's  salary  is  increased  to  $2500  ? 

37.  A  and  B  unite  in  conducting  a  summer  hotel,  on  the  follow- 
ing basis  :  1.  Each  is  to  receive  interest  at  the  rate  of  §%  per 
annum  on  his  investment ;  2.  A  is  to  receive  a  salary  of  $1000 
and  B  of  $800  for  the  season  ;  3.  The  profit  or  loss  of  the  general 
business  is  to  be  divided  in  the  proportion  of  A  f ,  B  \  ;  the  profit 
or  loss  of  the  livery  business  attached  thereto  in  the  proportion  of 
A  \,  B  f  ;  the  profit  or  loss  of  the  bathing  business  in  the  propor- 
tion of  A  £,  B  %.  A  invests  an  average  of  $10150  for  four  months, 
andB  an  average  of  $6750  for  the  same  time.  At  the  close  of  the 
business  the  accounts  showing  loss  and  gain  stand  as  follows  : 

Outgo.          HOTEL.       Income.  Outgo.      LIVERY.    Income. 

15150.75  |  25175.19         1592.75  |  3279.50 

Outgo.   BATHING.    Income. 


759.12     I     1275.30 

There  is  besides  an  item  of  service  amounting  to  $375,  which 
at  the  time  could  not  be  easily  apportioned  in  the  charges,  and 
which  does  not  appear  in  the  above  outgoes.  It  is  agreed  that 
this  item,  as  also  the  sums  severally  due  the  partners  for  interest 
and  salaries,  shall  be  charged  to  the  several  departments  of  the 
business  in  proportion  to  the  net  gains.  There  is,  also,  an  inven- 
tory in  the  livery  business  amounting  to  $429.33.  How  much 
gain  from  all  sources  will  each  partner  get  out  of  the  business  ? 

38.  A,  B,  and  0  are  equal  partners  in  a  mill,  each  to  receive 
6%  per  annum  interest  on  his  average  investment.  C  is  to  super- 
intend the  business  and  receive  therefor  a  yearly  salary  of  $3000; 
B  keeps  a  store  at  which  the  operatives  trade,  and  is  to  pay  to  A 
and  C  5%  on  sales  to  operatives.  A  negotiates  the  products  of  the 
mill,  for  which  he  is  allowed  10^  on  the  net  profits  as  existing 
before  his  percentage  is  taken.  A's  average  investment  for  the 
year  is  $9750  ;  B's  $5750  ;  C's  $5000.  Leaving  out  the  interest, 
salary  and  percentages,  the  net  gain  for  the  year  is  $15000.  B's 
sales  to  operatives  amount  to  $1575.  What  share  of  the  $15000  is 
each  partner  entitled  to  ? 


NATIONAL    BANKS. 


640.  A  National  Bank  is  a  bank  organized  under  the  laws 
of,  and  chartered  by,  the  United  States. 

1.  National  banks  are  authorized  to  discount  and  negotiate  notes,  drafts, 
etc. ;  to  receive  deposits ;  to  buy  and  sell  exchange,  coin  and  bullion ;  to  loan 
money  on  personal  security ;  and  to  issue  circulating  notes. 

2.  National  banks  are  prohibited  from  making  loans  on  real  estate  or  on 
security  of  their  own  shares  of  capital  except  to  secure  debts  previously  con- 
tracted. 

Real  Estate  purchased  or  mortgaged  to  secure  a  pre-existing  debt  shall  not 
be  held  for  a  longer  period  than  five  years. 

They  are  also  prohibited  from  making  loans  to  one  person  or  association, 
excepting  on  business  paper  representing  actually  existing  value  as  security, 
in  excess  of  one-tenth  of  the  capital  of  the  bank. 

3.  The  stockholders  of  a  national  bank  are  individually  liable  (equally 
and  ratably,  and  not  one  for  another)  for  an  amount  equal  to  the  par  value  of 
the  capital  stock  held  by  them. 

641.  Circulation.     Upon  a  deposit  of  registered  bonds,  the 
association  making  the  same  shall  be  entitled  to  receive  from  the 
Comptroller  of  the  Currency  circulating  notes  of  different  denomi- 
nations (12O)  equal  in  amount  to  90$  of  the  current  market  value, 
not  exceeding  par,  of  the  United  States  bonds  so  transferred  and 
delivered,  and  at  no  time  shall  the  total  amount  of  such  notes 
issued  to  any  association  exceed  90%  of  the  amount  at  such  time 
actually  paid  in  of  its  capital  stock. 

1.  Any  national  bank  desiring  to  decrease  its  circulation,  in  whole  or  in 
part,  may  deposit  lawful  money  (specie  or  legal  tenders)  with  the  Treasurer 
of  the  United  States  in  sums  of  not  less  than  $9,000,  and  withdraw  a  propor- 
tionate amount  of  bonds  held  as  security  for  such  notes. 

No  national  bank  which  makes  any  deposit  of  lawful  money  in  order  to 
withdraw  its  circulating  notes,  shall  be  entitled  to  receive  any  increase  of  its 
circulation  for  the  period  of  six  months  from  the  time  it  made  such  deposit. 
Not  more  than  $3,000,000  shall  be  deposited  during  any  calendar  month  for 
this  purpose. 

2.  The  State  bank  circulation  wholly  ceased  after  Congress  had  imposed 
a  penalty  of  10^  in  the  form  of  a  tax  every  time  it  should  be  issued.     This  act 
took  effect  Aug.  1,  1866. 


Art.  642.]  ,  NATIONAL     BANKS.  301 

642.  Redemption. — The  circulating  notes  of  national  banks 
are  redeemed  in  lawful  money  by  the  banks  which  issued  them 
and  by  the  Treasurer  of  the  United  States  at  Washington,  D.  C. 

1.  Every  national  bank  shall,  at  all  times,  keep  and  have  on  deposit  in  the 
Treasury  of  the  United  States  in  lawful  money  of  the  United  States,  a  sum 
equal  to  5^  of  its  circulation,  to  be  held  and  used  for  the  redemption  of  such 
circulation. 

2.  All  national  banks  which  go  into  voluntary  liquidation  shall,  within 
six  months  thereafter,  deposit  in  the  Treasury  an  amount  of  lawful  money 
equal  to  the  amount  of  their  circulating  notes  outstanding.     The  law  also 
requires  that  full  provision  shall  be  made  for  the  redemption  of  the  circulat- 
ing notes  of  any  insolvent  bank  before  a  dividend  is  made  to  its  creditors. 
Thus  it  will  be  seen  that  no  association  can  close  up  its  business  without  first 
providing  for  the  payment  of  all  its  circulating  notes,  and  that  the  amount 
deposited  for  their  redemption  must  remain  in  the  Treasury  until  the  last 
outstanding  note  shall  have  been  presented.     It  is  therefore  plain  that  the 
government,  and  not  the  bank,   receives  all  the  benefit  arising  from  lost  or 
unredeemed  circulating  notes. 

643.  Reserve. — The  national  banks  in  the  reserve  cities* 
are  required  by  law  to  hold  a  lawful  money  reserve  of  25$  of 
their  deposits  ;  all  other  national  banks  15$.     The  excess  above 
legal  requirements  is  called  "  surplus  reserve." 

644.  Surplus  Fund. — The  law  provides  that  a  surplus  fund 
shall  be  accumulated,,  by  setting  aside,   before  the  usual  semi- 
annual dividend  is  declared,  one-tenth  part  of  the  net  profits  of 
the   bank  for  the   preceding   half-year,  until  the  surplus  fund 
shall  amount  to  20$  of  its  capital  stock. 

EXAM  PLES. 

645.  1.  The  impairment  of  the  capital  stock  ($300000)  of  an 
insolvent  national  bank  was  $216000.     What  was  the  rate  per 
cent,  of  the  assessment  made  upon  the  stockholders  for  the  pur- 
pose of  making  good  the  deficiency  (64O,  3)  ?    How  much  was 
Mr.  A.  obliged  to  pay,  who  owned  80  shares  ? 

2.  What  amount  of  bank  notes  is  issued  to  a  national  bank 
that  deposits  $780000  in  U.  S.  bonds  to  secure  circulation  (641)  ? 
How  much  is  its  redemption  fund  (642,  1)  ? 

*  The  reserve  cities  are  New  York,  Boston,  Philadelphia,  Baltimore,  Albany,  Pitts- 
burgh, Washington,  New  Orleans,  Louisville,  Cincinnati,  Cleveland,  Chicago,  Detroit,  Mil- 
waukee, Saint  Louie,  and  San  Francisco. 


302  NATIONAL     BANKS.  [Art.  645. 

3.  A  national  bank,  desiring  to  reduce  its  circulation,,  deposits 
with  the  Treasurer  of  the  United  States  $27000  in  legal-tenders, 
and  sells  the  bonds  withdrawn  (641)  in  the  market  at  118J. 
What  were  the  proceeds  ? 

4.  The  circulation  of  a  national   bank   having  a  capital  of 
$150000  is  $57600  ;  what  is  the  remaining  amount  of  circulation 
which  it  may  call  for  by  depositing  the  necessary  amount  of  bonds 
(641)  ?     What  is  the  par  value  of  the  bonds  now  on  deposit  ? 
What  additional  amount  of  bonds  must  the  bank  deposit  if  the 
circulation  is  increased  to  the  maximum  ? 

5.  How  much  is  the  redemption  fund  of  a  bank  whose  circula- 
ation  is  $427500  ?    What  is  the  amount  of  bonds  on  deposit  to 
secure  its  circulation  ? 

6.  The   New   York   associated  banks,    Mar.    25,    1882,    held 
$58,602,100  in  specie  and  $16,150,900  in   legal-tenders.     Their 
deposits  on  the  same   date  were   $285,659,600.     What  was   the 
excess  of  reserve  (643)  above  legal  requirements  ? 

7.  Oct.  1,  1881,  the  national  banks  of  Boston  had  $8,286,182 
in  specie,  $3,457,379  in  legal-tenders,  $75,000  in  U.  S.  certificates 
of  deposit,  $11,735,499  due  from  reserve  agents,  and  a  redemption 
fund  with  U.  S.  Treasurer  of  $1,603,628.     What  was  the  ratio  ol 
the  reserve   to   the   deposits,   which   were   $95,776,386  ?     What 
amount  of  reserve  was  required  ?    What  was  the  surplus  reserve  ? 

8.  What  amount  of  reserve  was  required  by  the  national  banks 
of  the  State  of  Maine,  their  deposits  being  $9,558,878  ? 

9.  The  net  earnings  of  a  bank,  whose  surplus  (644)  is  less 
than  20%  of  its  capital  ($300000),  are  $10475.38.     What  amount 
must  be  carried  to  the  surplus  account,  and  what  are  the  undivided 
profits  after  declaring  a  dividend  of  3%  ? 

10.  What  is  the  semi-annual  tax*  at  \%  upon  a  national  bank 
whose  average  circulation  is  $462,730  (641)? 

11.  A  bank  having  a  capital  of  $250,000,  and  a  surplus  of 
$50,000,  for  a  period  of  six  months,  earned  $58693,  and  declared 
a  dividend  of  $30000.     What  was  the  rate  of  the  dividend  ?     The 
dividend  is  what  %  of  the  capital  and  surplus  ?     The  net  earnings 
are  what  %  of  the  capital  and  surplus  ? 

12.  The  average  daily  exchanges  at  the  New  York  Clearing 
House  for  1886  were  $109,000,000,  and  the  average  daily  balances 
$5,000,600.     The  balances  were  what  %  of  the  exchanges  ? 

»  The  tax  upon  capital  and  deposits  was  repealed  by  Act  of  March  3, 1883. 


SAVINGS      BANKS. 


646.  Savings  Banks  are  institutions  for  the  deposit  and 
safe  keeping  of  small  sums  of  money.      They  are  designed   to 
encourage  thrift  and  economy  among  the  working  classes. 

647.  Interest  is  usually  declared  Jan.  1st  and  July  1st  of 
each  year,  and  when  declared  is  carried  at  once  to  the  credit  of 
each  depositor  on    the  books  of  the  bank,,  when  it  stands  as  a 
deposit,  and  is  entitled  to  interest  the  same  as  any  other  deposit. 
Savings  banks,  therefore,  pay  compound  interest. 

No  interest  is  allowed  on  the  fractional  parts  of  a  dollar,  nor  is 
any  interest  allowed  on  any  sum  withdrawn  previous  to  the  first 
day  of  January  or  July,  for  the  period  which  may  have  elapsed 
since  the  last  dividend. 

648.  Deposits  are  practically  payable  on  demand,  though 
the  right  to  require  a  notice  of  60  or  90  days  is  reserved. 

In  some  savings  banks,  deposits  commence  to  draw  interest 
Jan.  1st,  April  1st,  July  1st,  Oct.  1st ;  in  others,  deposits  made 
on  or  before  the  first  of  any  month  draw  interest  from  the  first 
days  of  those  months  respectively. 

NOTE. — In  certain  savings  banks,  money  must  be  on  deposit  at  least  three 
months  hef  ore  it  will  be  entitled  to  any  interest.  In  a  bank  that  has  the  above 
by-law  and  that  pays  interest  from  the  first  of  each  month,  money  deposited 
on  or  before  May  1  would  draw  interest  for  8  months  on  the  following  Jan.  1, 
and  deposits  made  on  or  before  June  1  would  draw  interest  for  7  months, 
Jan.  1.  Deposits  made  in  October  and  November  would  be  treated  in  a 
similar  manner  on  the  following  July  1. 

649.  According  to  the  laws  of  the  State  of  New  York, 

No  person  shall  have  a  deposit  larger  than  the  sum  of  three  thousand 
dollars,  exclusive  of  accrued  interest,  unless  such  deposit  was  made  prior  to 
the  passage  of  the  act  (May  17,  1875),  or  pursuant  to  the  order  of  a  court 
of  record,  or  of  a  surrogate. 


304  SAVINGS      BANKS.  [Art.  640. 

Savings  banks  are  restricted  to  5$  per  annum  regular  interest  or  divi- 
dend. They  must,  however,  declare  an  extra  dividend  at  least  once  in  three 
years,  when  their  surplus  earnings  amount  to  15 %  of  their  deposits. 

Savings  banks  are  allowed  to  pay  interest  on  all  sums  deposited  during 
the  first  ten  days  of  January  and  July,  and  the  first  three  days  of  April  and 
October  from  the  first  of  those  months  respectively. 

EXAMPLES. 

65O.  Perform  the  following  examples  according  to  both 
methods  mentioned  in  Art.  648.  Where  no  rate  is  mentioned, 
4%  is  understood. 

1.  A  person  deposited  Dec.  30,  1881,  $150;  Feb.  20,  1882, 
$40  ;  April  1, 1882,  $120  ;  May  30,  1882,  $60.     What  amount  was 
due  July  1,  1882,  nothing  having  been  withdrawn  ? 

ANALYSIS. — If  interest  begin  on  the  first  of  each  quarter,  the  first  deposit, 
$150,  will  draw  interest  from  Jan.  1,  or  for  6  mo. ;  the  second  and  third 
deposits,  $160  ($40 +$120),  will  draw  interest  from  April  1,  or  for  3  mo. ;  the 
last  deposit,  made  May  30,  will  draw  no  interest  July  1. 

4  %  per  annum  is  2%  for  6  months  and  1  %  for  3  months. 

If  interest  begin  on  the  first  of  each  month,  the  first  deposit,  $150,  will 
draw  interest  from  Jan.  1,  or  for  6  mo. ;  the  second  deposit,  $40,  made  Feb. 
20,  will  draw  interest  from  March  1,  or  for  4  mo. ;  the  third  deposit,  $120, 
made  April  1,  will  draw  interest  from  April  1,  or  3  mo.;  the  fourth  deposit, 
$160,  made  May  30,  will  draw  interest  from  June  1,  or  for  1  mo. 

2.  The  following  deposits  were  made  in  a  savings  bank  :  July 
1,  1881,  $100 ;  July  16,  $40  ;  Aug.  1,  $75 ;  Aug.  29,  $45 ;  Sept. 
30,  $75  ;  Oct.  28,  $200  ;  Nov.  25,  $30  ;  Dec.  31,  $100.     What  was 
due  Jan.  1,  1882  ? 

NOTE. — Balance  the  following  accounts  Jan.  1  and  July  1  of  each  year. 

8.  How  much  interest  was  due  on  the  following  account  July 
1, 1883  ?  Deposits,  Oct.  1,  1881,  $200 ;  Dec.  31, 1881,  $160  ;  Mar. 
24,  1883,  $100. 

4.  Mr.  A.  made  the  following  deposits  in  a  savings  bank  :  Jan. 
1,  1879,  $100  ;  May  1,  1879,  $140;  June  30,  1879,  $40;  Oct.  1, 
1879,  $60 ;  Feb.  28,  1880,  $120 ;  June  30,  1880,  $45  ;  Aug.  29, 
1881,  $200.     What  was  the  balance  due  Jan.  1,  1882  ? 

5.  What  is  the  balance  of  the  following  account  July  1,  1879, 
interest  being  reckoned  at  §%  until  July  1,  1877,  and  at  5%  there- 
after :  Deposits,  Oct.  14,  1876,  $200 ;  Mar.  30,  1878,  $135  ;  April 
1,  1879,  $90  ? 


Art.  650.] 


SAVINGS     BANKS. 


305 


6.  Balance  the  following  account  July  1.  Balance  due  Jan.  1, 
8103.  Deposits,  Jan.  28,  $40 ;  Mar.  30,  $125  ;  May  26,  $80. 
Drafts,  Feb.  20,  $20;  April  18,  $15  ;  May  3,  $25;  June  16,  $100. 


Date, 

Deposits. 

Drafts. 

Jan.       1, 

103 

28, 

^0 

Feb.     20, 

20 

n 

Mar.    30, 

m 

Apr,     18, 

** 

z$ 

May       3, 

65 

n 

26, 

n 

June    16, 

*00 

ANALYSIS. — In  order  to  determine  the 
amounts  that  are  entitled  to  interest, 
arrange  the  account  in  the  following  form, 
and  deduct  the  drafts  from  the  last  deposits 
made,  by  the  system  of  cancellation  indi- 
cated below.  The  draft  of  $20  made  Feb. 
20  is  deducted  from  the  deposit,  $40,  of 
Jan.  28,  leaving  $20.  The  drafts  made 
April  18,  $15,  and  May  8,  $25,  are  de- 
ducted from  the  deposit  of  March  30, 
leaving  $85.  The-  draft  of  June  16,  $1QO, 
cancels  all  of  the  deposit  of  May  26,  and 
$20  of  the  deposit  of  March  30,  leaving 

$65  ($85— $20).      The  net  deposits  are  as  follows  :  Jan.  1,  $103  ;  Jan.  28, 
$20;  March  30,  $65. 

If  interest  commence  the  first  of  each  quarter,  the  several  amounts  will 
draw  interest  as  follows  :  $103  from  Jan.  1,  or  6  mo. ;  $20,  deposited  Jan.  28, 
and  $65  deposited  Mar.  30,  making  $85  from  April  1,  or  3  mo. 

If  interest  commence  the  first  of  each  month,  the  several  amounts  will 
draw  interest  as  follows  :  $103  from  Jan  1,  or  6  mo.  ;  $20,  deposited  Jan.  28, 
from  Feb.  1,  or  5  mo.  ;  $65,  deposited  Mar.  30,  from  April  1,  or  3  mo. 

7.  What   is   the   balance  of  the   following  account   July  1  ? 
Balance  due  Jan.  1,  $30 ;  deposits,  Feb.  16,  $50  ;  Apr.   1,  $185. 
Drafts,  Mar.  12,  $60  ;  May  10,  $50 ;  June  20,  $60. 

8.  Balance  the  following  Jan.  1,  1881.     Deposits,  July  1, 1880, 
$300 ;  Aug.  1,  $150  ;  Sept.  27,  $60  ;  Oct.  12,  $325.     Drafts,  July 
16,  $150;  Sept.  1,  $150  ;  Nov.  17,  $70;  Dec.  18,  $140. 

9.  Balance  the  following  account  July  1.     Balance  due  Jan. 
1,  $364.48.     Deposits,  Jan.  24,  $50  ;  Feb.  16,  $80  ;  Apr.  30,  $40 ; 
June  28,  $100.     Drafts,  Mar.  30,  $75  ;  May  19,  $10. 

10.  What  was  due  July  1,  1882,  on  the  following  pass-book  ? 

Dr.        FRANKLIN  SAVINGS  BANK  in  account  with  A.  C.  LOBECK.         Cr. 


1881. 

! 

1881. 

1 

Jan.    1 

Four  Hundred  Dollars. 

400 

Aug.  1 

Two  Hundred  Dollars. 

300 

Mar.  15 

Ninety  Dollars. 

90 

1882. 

1881. 

Interest  to  July. 

* 

** 

Jan.  16 

One     Hundred      and 

Sept.  16 

Two  Hundred  Dollars. 

200 

Sixty  dollars. 

160 

1882. 

Interest  to  January. 

* 

** 

June  1 

Eighty  Dollars. 

80 

Feb.  27 

Two  Hundred  and  Sixty  Dollars. 

260 

Mar.    8 

One  Hundred  Dollars. 

100 

LIFE     INSURANCE. 


651.  Life  Insurance  is  a  contract  by  which  a  company  (the 
insurer),  in  consideration  of  certain  payments,  agrees  to  pay  to  the 
heirs  of  a  person,  when  he  dies,  or  to  himself,  if  living  at  a  specified 
age,  a  certain  sum  of  money. 

Life  Insurance  Companies  may  be  classified  according  to  principles  of 
organization  the  same  as  Fire  Insurance  Companies  (5259). 

652.  The  principal  kinds  of  policies  issued  by  Life  Insurance 
Companies  are  the  following :   Ordinary  Life,  Limited  Pay- 
ment Life,  Endowment,  and  Annuity. 

Tontine  Investment,  Reserve  Endowment,  Semi-Tontine,  Semi-Endow- 
ment, Yearly  Renewable,  and  other  special  policies  are  issued  by  some  com- 
panies. 

653.  Ordinary  Life  Policies.— On  this  kind  of  policy,  a 
certain  premium  is  to  be  paid  every  year  until  the  death  of  the 
insured,  when  the  policy  becomes  payable  to  the  persons  named  in 
the  policy  as  the  beneficiaries. 

654.  Limited  Payment  Life  Policies. — On  a  policy  of 
this  kind,  premiums  are  paid  annually  for  a  certain  number  of 
years  fixed  upon  at  the  time  of  insuring — or,  until  the  death  of 
the  insured,  should  that  occur  prior  to  the  end  of  the  selected 
period.     The  policy  is  payable  on  the  death  of  the  insured. 

These  policies  are  issued  with  single  payments,  or  with  5,  10,  15,  20,  or  25 
annual  payments. 

655.  Endowment  Policies. — An  Endowment  Policy  pro- 
vides (1)  insurance  during  a  stipulated  period,  payable  at  the 
death  of  the  insured  should  he  die  within  the  period ;  and  (2)  an 
endowment,  of  the  same  amount  as  the  policy,  payable  at  the 
end  of  the  period  if  the  insured  survive  until  that  time. 

These  policies  are  issued  for  endowment  periods  of  10,  15,  20,  25,  30,  or  35 
years,  and  may  be  paid  up  by  a  single  payment,  by  annual  premiums  during 
the  endowment  period,  or  by  5  or  10  annual  payments. 


Art.  656.]  LIFE     INSURANCE.  307 

656.  Annuity  Policies.  —  An  Annuity  Policy  secures  to  the 
holder  the  payment  of  a  certain  sum  of  money  every  year  during 
his  life-time.     It  is  secured  by  a  single  cash  payment. 

657.  The  Reserve  of  life  insurance  policies  is  the  present 
value  of  the  amount  to  be  paid  at  death  less  the  present  value  of 
all  the  net  premiums  to  be  paid  in  the  future. 

658.  The  Reserve  Fund  of  a  Life  Insurance  Company  is 
that  sum  in  hand  which,  invested  at  a  given  rate  of  interest  to- 
gether with  future  premiums  on  existing  policies,  should  be  suf- 
ficient to  meet  all  obligations  as  they  become  due.     It  is  the  sum 
of  the  separate  reserves  of  the  several  policies  outstanding. 

The  legal  rate  for  the  reserve  fund  according  to  the  laws  of  the  State  of 
New  York,  is  4%ft>  ;  of  Massachusetts,  4$>. 


659.  A  Non-Forfeiting  Policy  is  one  which  does  not  be- 
come void  on  account  of  non-payment  of  premiums. 

1.  According  to  the  laws  of  the  State  of  New  York,  after  three  full  annual 
premiums  have  been  paid,  the  legal  reserve  of  the  policy,  calculated  at  the 
date  of  the  failure  to  make  the  payments,  shall,  on  surrender  of  the  policy 
within  six  months  after  such  lapse,  be  applied  as  a  single  payment  at  the 
published  rates  of  the  company  in  either  of  two  ways,  at  the  option  of  the 
assured.     (1)  To  the  continuance  of  the  full  amount  of  the  insurance  so  long 
as  such  single  premium  will  purchase  term  insurance  for  that  amount,  or  (2) 
to  the  purchase  of  a  non-participating  paid-up  policy. 

2.  According  to  the  Massachusetts  limited  forfeiture   law  of  1880,  after 
two  full  annual  premiums  have  been  paid,  and  without  any  action  on  the  part 
of  the  assured,  the  net  value  (Massachusetts  standard)  of  the  policy  less  a  sur- 
render charge  of  8%  of  the  present  value  of  the  future  premiums  which  the 
policy  is  exposed  to  pay  in  case  of  its  continuance,  shall  be  applied  as  a  single 
payment  to  the  purchase  of  paid-up  insurance. 

3.  Certain  companies  voluntarily  apply  all  credited  dividends  to  the  con- 
tinuance of  the  insurance  ;  others  voluntarily  apply  the  legal  reserve  to  the 
purchase  of  term  insurance  at  the  regular  rates. 

4.  In  some  companies,  all  limited  payment  life  policies  and  all  endow- 
ment policies,  after  premiums  for  three  (or  two)  years  have  been  paid  and  the 
original  policy  is  surrendered  within  a  certain  time,  provide  for  paid-up  assur- 
ance for  as  many  parts  (tenths,  fifteenths,  twentieths,  etc.,  as  the  case  may 
be),  of  the  original  amount  assured,  as  there  shall  have  been  complete  annual 
premiums  received  in  cash  by  the  Company. 

660.  The  Surrender  Value  of  a  policy  is  the  amount  of 
cash  which  the  company  will  pay  the  holder  on  the  surrender  of 
the  policy.     It  is  the  legal  reserve  less  a  certain  per  cent,  for 
expenses. 


308 


LIFE     INSURANCE. 


[Art.  661, 


TABLE  OF  KATES. 
661.  Annual  premium  for  an  Insurance  of  $1,000,  with  profits. 


LIFE  POLICIES. 
Payable  at  Death,  only. 

ENDOWMENT  POLICIES. 

Payable  as  Indicated,  or  at  Death,  if  Prior. 

AGE. 

ANNUAL  PAYMENTS. 

AGE. 

In 
10 
Years. 

In 
15 

Years. 

In 
20 
Years. 

AGE. 

For  Life. 

10  Years. 

15  Years. 

20  Years. 

25 

$19  89 

$42  56 

$32  24 

$27  39 

25 

$103  91 

$66  02 

$47  68 

25 

26 

20  40 

43  37 

32  97 

27  93 

26 

104  03 

66  15 

47  82 

26 

27 

20  93 

44  22 

33  62 

28  50 

27 

104  16 

66  29 

47  98 

27 

28 

21  48 

45  10 

34  31 

29  09 

28 

104  29 

66  44 

48  15 

28 

29 

22  07 

46  02 

35  02 

29  71 

29 

104  43 

66  60 

48  33 

29 

30 

22  70 

46  97 

35  76 

30  36 

30 

104  58 

66  77 

48  53 

30 

31 

23  35 

47  98 

36  54 

31  03 

31 

104  75 

66  96 

48  74 

31 

32 

24  05 

49  02 

37  35 

31  74 

32 

104  92 

67  16 

48  97 

32 

33 

24  78 

50  10 

t  38  20 

32  48 

33 

105  11 

67  36 

49  22 

33 

34 

25  56 

51  22 

39  09 

33  26 

34 

105  31 

67  60 

49  49 

34 

35 

26  38 

52  40 

40  01 

34  08 

35 

105  53 

67  85 

49  79 

35 

36 

27  25 

53  63 

40  98 

34  93 

36 

105  75 

68  12 

50  11 

36 

37 

28  17 

54  91 

42  00 

35  83 

37 

106  00 

68  41 

50  47 

37 

38 

29  15 

56  24 

43  06 

36  78 

38 

106  28 

68  73 

50  86 

38 

39 

30  19 

57  63 

44  17 

37  78 

39 

106  58 

69  09 

51  30 

39 

40 

31  30 

59  09 

45  33 

38  83 

40 

106  90 

69  49 

51  78 

40 

41 

32  47 

60  60 

46  56 

39  93 

41 

107  26 

69  92 

52  31 

41 

42 

33  72 

62  19 

47  84 

41  10 

42 

107  65 

70  40 

52  89 

42 

43 

35  05 

63  84 

49  19 

42  34 

43 

108  08 

70  92 

53  54 

43 

44 

38  46 

65  57 

50  61 

43  64 

44 

108  55 

71  50 

54  25 

44 

45 

37  97 

67  37 

52  11 

45  03 

45 

109  07 

72  14 

55  04 

45 

46 

39  58 

69  26 

53  68 

46  50 

46 

109  65 

72  86 

55  91 

46 

47 

41  30 

71  25 

55  35 

48  07 

47 

110  30 

73  66 

56  89 

47 

48 

43  13 

73  32 

57  10 

49  73 

48 

111  01 

74  54 

57  96 

48 

49 

45  09 

75  49 

58  95 

51  50 

49 

111  81 

75  51 

59  15 

49 

50 

47  18 

77  77 

60  91 

53  38 

50 

112  68 

76  59 

60  45 

50 

1.  The  above  table  represents  the  maximum  rates  of  the  leading  New 
York  companies.     Surplus  premiums  or  dividends  are  returned  annually  com- 
mencing at  the  payment  of  the  second  premium. 

2.  Policies  which  do  not  share  in  the  dividends  of  the  company,  are  issued 
at  fixed  rates  15  to  20$  less  than  the  above. 

3.  The  above  rates  are  for  annual  payments  only.     To  obtain  semi-annual 
payments,  add  k%  and  divide  by  2.     To  obtain  quarterly  payments,  add  6  % 
and  divide  by  4. 


Art.  662.]  LIFE  INSURANCE.  309 


EXAMPLES. 

662.     1.  Find  the  amount  of  premium  for  an  ordinary  life 
policy  (653,  661)  of  $5000,  issued  to  a  person  35  years  of  age. 

2.  What  is  the  first  annual  premium  of  a  life  policy  of  $6000, 
issued  to  a  person   30   years   old,  $1.00   being   charged   for   the 
policy  ? 

NOTE. — The  policy  fee  is  added  to  the  first  premium  only. 

3.  Find  the  annual  premium  for  a   20-payment   life   policy 
(654,  661)  of  $4000,  issued  to  a  person  28  years  old. 

4.  What  annual  premium  must  be  paid  for  a  20-year  endow- 
ment policy  (655)  of  $8000,  age  of  the  insured  at  nearest  birth- 
day, 40  years  ?     If  the  insured  dies  during  the  tenth  year,  how 
much  more  would  have  been  paid  than  if  he  had  been  insured  on 
the  ordinary  life  plan  ? 

5.  What  is  the  average  daily  cost  of  a  life  policy  for  $1000,  no 
allowance  being  made  for  probable  dividends,  insurance  commenc- 
ing at  age  25  ?    At  35  ?     At  45  ? 

6.  How  much  must  a  person,  aged  35,  lay  aside  weekly  to 
secure  a  life  policy  of  $1000,  payable  in  20  annual  payments  ? 

7.  When  40  years  old,  a  person  took  out  a  20-year  endowment 
policy  of  $10000.     He  survived   the   endowment   period.     How 
much  less  did  he  receive  than  he  paid  as  premiums,  not  reckon- 
ing interest  ? 

8.  Mr.  A.  when  26  years  old  took  out  an  ordinary  life  policy 
of  $20000.     He  died  aged  41  years  2  months.     How  much  more 
did  his  heirs  receive  than  had  been  paid  premiums,  no  allowance 
being  made  for  interest  ? 

9.  In   the  above  example,  supposing  money  to  be  worth   $% 
(simple  interest),  what  was  the  net  gain  of  the  above  insurance  ? 

10.  The  annual  premium,  without  profits,  on  a  life  policy  of 
$10000  at  age  35  is  $222.     How  much  would  it  be  necessary  to 
invest  at  6%  interest  to  secure  the  payment  of  the  annual  pre- 
mium ?     How  much  would  the  insured  leave  his  family   at   his 
death  ? 

11.  A  gentleman,  age  30,  insures  his  life  for  $20000,  ordinary 
life  plan.     How  much  must  he  place  in  trust  so  that  the  interest 
at  6%  will  be  sufficient  to  pay  the  premiums  on  the  policy  ?     At 
his  death,  how  much  does  he  leave  his  family? 


310  LIFE      INSURANCE.  [Art.  6G2. 

12.  If  a  man  32  years  old  takes  out  a  life  policy  for  $5000  and 
dies  just  before  reaching  the  age  of  40  years,  how  much  less  will 
his  total  payments  be  than  if  he  had  taken  a  20-year  endowment 
policy  for  the  same  amount  ? 

IS.  Mr.  C.  when  25  years  of  age  secured  a  20-year  endowment 
policy  of  $6000  ;  when  he  was  30  years  of  age,  he  obtained  an 
ordinary  life  policy  of  $4000  ;  when  35  years  of  age,  he  took  out 
a  20-payment  life  policy  of  $10000.  What  was  the  total  annual 
premium  after  taking  the  last  policy  ? 

14.  Suppose  Mr.  C.  had  died  at  the  age  of  40^  years,  how  much 
more  would  his  heirs  receive  than  had  been  paid  as  premiums  ? 

15.  A  single  premium  for  an  assurance  of  $1000, without  profits, 
for  a  person  32  years  old,  is  $300.     What  would  be  the  excess  of 
the  assurance  over  the  amount  produced  by  placing  the  money  at 
compound  interest  (483)  at  4$,  supposing  the  insured  to  live  20 
years  ?     30  years  ?    What  would  be  the  excess  of  the  sum  produced 
by  the  money  at  interest  at  5$,  over  the  assurance  in  30  years  ? 

16.  Mr.  B.,  age  40,  has  $10000  at  interest  at  6$,  which  he 
intends  to  leave  his  family.     What  will  this  amount  to  at  com- 
pound interest  (483)  in  25  years  at  6%  ?     How  much  will  he 
leave  his  family  if  he  takes  out  a  life  policy  and  pays  the  premium 
with  the  intesest  on  his  investment  of  $10000  ? 

17.  Mr.  A.,  aged  30,  secures   an  ordinary  life  policy,  annual 
premium  $100.     How  much  more  would  his  heirs  receive  from 
the  insurance  company  than  from  the  money  at  compound  in- 
terest (484)  at  5#,  should  he  die  at  the  age  of  32  ?     Of  40  ?     Of 
50  ?    At  about  what  age  would  the  amount  received  from  the 
money  at  interest  exceed  the  assurance  ? 

18.  What  is  the  semi-annual  premium  (661,  3)  on  a  20-year 
endowment  policy  for  $6000,  age  32  ?     The  quarterly  premium  ? 

19.  Mr.  A.,  who  will  be  35  years  of  age  July  1,  takes  out  Apr.  1 
a  20-payment  life  policy  for  $10000,  premium  payable  semi-annu- 
ally.     Mr.  B. ,  of  the  same  age,  takes  out  Apr.  1  the  same  kind  of 
policy  for  $5000,  and  Oct.  1,  another  policy  of  the  same  kind  for 
$5000,  premium  payable  annually.     How  much  less  does  Mr.  B. 
pay  as  premium  each  year  than  Mr.  A  ?     (661,  3.) 

20.  An  ordinary  life  policy  issued  at  age  35  for  $10000  has,  at 
age  45,  a  4%  reserve  of  $1262.60.     How  much  non-participating 
paid-up  insurance  will  this  amount  purchase,  the  single  premium 
rate  per  $1000  at  age  45  being  $475.44  ? 


Art.  663.]  REVIEW     EXAMPLES.  311 


REVIEW  EXAMPLES. 

663.  1.  Add  17J,  28f,  36-| ,  44f ,  89T\,  and  76  J  ;  multiply  the 
sum  by  87 ;  subtract  1022|£  from  the  product ;  and  divide  the 
remainder  by  234f . 

2.  Divide  eighty-three,  and  seventy-five  hundredths  by  one  hun- 
dred twenty-five  ten-thousandths  ;  add  to  the  quotient  sixty-eight, 
and  six  hundred  twenty-five  thousandths  ;  and  multiply  the  sum  by 
three,  and  two-tenths. 

3.  How  many  minutes  in  the  month  of  February,  1900  ? 

4.  Find  the  cost  of  7312  pounds  of  meal  at  $2.25  per  civt. 

5.  The  difference  in  the  local  time  of  two  places  is  1  hr.  7  min. 
13  sec.;  what  is  the  difference  in  longitude  ? 

6.  Find  the  number  of  square  yards  of  paving  in  a  street,  3000 
ft.  long  and  50  ft.  wide. 

7.  What  is  the  charge  for  packing,  marking,  and  shipping  251 
bales  merchandise  at  6s.  6d.  per  bale  ? 

8.  If  46  T.  12  cwt.  of  coal  are  worth  $174.75,  what  is  the  value 
of  37  T.  8  cwt.  ? 

9.  How  many  square  yards  of  linoleum  would  cover  a  floor 
22ft.  6  in.  by  15  ft.  4  in.?    Find  its  value  at  63^  per  sq.  yd. 

10.  What  is  the  freight  of  5  T.  9  cwt.  2  qr.  8  lb.,  at  70  shillings 
per  ton  (2240  lb.)? 

11.  Bought  280  cords  of  hard  wood,  at  $6. 75,  and  790  cords  of 
soft  wood,  at  $3.62|-  per  cord.     Also,  750  bushels  of  corn,  at  62 J 
cents,  and  925  bushels  of  oats,  at  37 \  cents  per  bushel.     What  was 
paid  for  the  whole,  and  what  was  the  average  price  of  wood  per 
cord,  and  of  grain  per  bushel  ? 

12.  Bought  on  contract  350  reams  of  foolscap  paper,  at  $3. 83| 
per  ream,  45J  reams  of  which  were  returned  as  unsuitable,  and  275 
reams  of  letter,  at  $2.67|-  per  ream,  37-|  reams  of   which  were 
rejected.     How  much  was  paid  for  the  remainder  ? 

13.  Feb.  26,  1879,  the  Nevada  Bank  of   San  Francisco  sold 
100,000  ounces  of  pure  silver  to  the  United  States,  at  $1.08|  per 
ounce.     At  this  rate,  what  is  the  intrinsic  gold  value  of  the  stand- 
ard silver  dollar  ? 

14.  What  is  the  value  of  45000  tons  of  steel  rails  at  97s.  6d. 
per  ton  ?     What  is  the  value  per  ton  in  U.  S.  money  ?     Of  total 
in  U.  S.  money  ? 


312  REVIEW    EXAMPLES.  [Art.  663. 

15.  What  will  be  the  cost  of  painting  the  walls  and  ceiling  of 
a  room,  whose  height,  length,  and  breadth  are  12  ft.  6  in.,  27  ft., 
and  20ft.,  respectively,  at  24  cents  per  square  yard  ? 

16.  What  is  the  total  cost  of  56123  bushels  oats  at  43  cents  per 
bushel,  and  41 114  bushels  corn  at  46  cents  per  bushel  ? 

17.  Find  the  total  freight  on  68  cu.Jt.  mdse.  at  35  shillings 
per  ton  (40  cu.  ft.},  and  123  cu.  ft.  at  40  shillings  per  ton,  plus 
10%  primage  on  each  item. 

18.  What  is  the  cost  of  250  ft.  3-ply  hose,  at  60  cts.  per  foot, 
less  30  and  10%,  and  5  sets  couplings  at  $1.50  each  ? 

19.  May  10,  A  buys  a  bill  of  goods  amounting  to  $5000  on 
the  following  terms  :  60  days,  or  \%  discount  in  30  days,  or  2% 
discount  in  10  days.     May  20,  he  makes  a  payment  of  $2000,  and 
June  9,  of  $2500.     How  much  would  be  due  July  9,  the  end1  of 
the  60  days'  credit  ? 

20.  Oct.  16,  B  bought  a  bill  of  merchandise  amounting  to 
$2000  on  the  following  terms  :  4  months,  or  5%  discount  in  30 
days,  or  6%  discount  in  10  days.     Oct.  26  he  made  a  payment  of 
$1000.     How  much  would  settle  the  bill  Nov.  15  ? 

21.  B  bought  a  bill  of  merchandise  May  16  amounting  to 
$3416.72  on  the  following  terms  :  4  mos.,  or  less  5%  30  days.     He 
paid  on  account  June  21  (6  days  after  the  expiration  of  the  30 
days)   $3000,  with  the  understanding  that  he  should  have  the 
benefit  of  the  discount  by  paying  interest  for  the  time  elapsed, 
at  6%  per  annum.     How  much  was  due  Sept.  16,  no  compound 
interest  being  reckoned  ? 

22.  A  commission  merchant  in  Chicago  sells  for  me  12  bales 
brown  sheeting,  each  bale  containing  800  yards,   at   7  cts.  per 
yard ;  pays  transportation  and  other  charges  amounting  to  $72  ; 
and  invests  the  proceeds  in  flour  at  $4.80  per  barrel.     If  he  charges 
%\%  for  selling  and  \\%  for  purchasing,  how  many  barrels  of  flour 
does  he  send  me  ? 

28.  A  of  Chicago,  sends  to  B  of  New  Orleans,  8000  bu.  of 
wheat  and  500  bbls.  of  flour  with  instructions  to  sell  it  and 
invest  the  proceeds  in  sugar.  B  pays  freight  and  cartage 
$3420;  sells  the  wheat  at  $1.60  per  bushel  and  the  flour  at  $5.25 
per  barrel ;  charges  2J%  commission  on  the  flour  and  \$ 
per  bushel  on  the  wheat.  How  many  pounds  of  sugar  are  pur- 
chased at  8J  cents  per  pound,  the  commission  for  purchasing 
being  3%  ? 


Art.  663.]  REVIEW    EXAMPLES.  313 

24.  Mr.  B.  purchased  3G150  pounds  of  hay  at  $16.50  per  ton, 
and  16438  pounds  of  oats  at  70  cents  per  bushel.    He  sold  the  hay 
at  a  gain  of  16$,  and  the  oats  at  a  loss  of  8$.     What  were  the 
proceeds  ? 

25.  If  I  purchase  two  building  lots  for  $3750  each,  and  sell 
one  for  J  more  than  it  cost,  and  the  other  for  33 \%  less,  what  is  the 
gain  or  loss  on  the  two  lots  ? 

26.  A  speculator  sells  two  farms  for  $6000  each ;  how  much 
does  he  gain  or  lose,  if  he  sells  one  for  20$  more  than  it  cost,  and 
the  other  for  -J  less  than  it  cost  ? 

27.  Bought  coal  by  the  long  ton  at  $3.64,  and  sold  by  the 
short  ton  at  $4.25.     What  was  the  gain  per  cent  ? 

28.  Mr.  A  oifered  to  sell  his  horse  for  12$  more  than  it  cost 
him,  but  afterward  sold  it  for  $504,  which  was  10$  less  than  his 
first  asking  price.     How  much  did  his  horse  cost  him  ? 

29.  Find  the  interest  of  $375.60  for  1  yr.  10  mo.  22  da.,  at 


SO.  Find  the  interest  of  $4128  for  8  mo.  26  da.,  at 

31.  What  is  2J$  of  £159  13s.  lOd.  ? 

82.  Find  the  date  of  maturity  and  the  net  proceeds  of  a  note 
for  $5000,  dated  May  16,  payable  4  months  after  date,  and  dis- 
counted July  21  at  6$. 

S3.  When  the  above  note  became  due,  its  maker  had  discount- 
ed at  6$  a  new  note,  payable  90  days  after  date,  whose  proceeds 
were  sufficient  to  pay  the  first  note.  What  was  the  face  of  the 
new  note  ? 

34>  Apr.  1,  a  merchant  buys  a  quantity  of  coffee  on  90  days' 
credit,  with  privilege  of  discounting  within  30  days  from  date  of 
purchase  at  the  rate  of  6$  per  annum  for  the  unexpired  time. 
Apr.  16,  he  makes  a  payment  of  $28000  on  account,  no  actual  in- 
voice having  been  rendered.  May  1,  he  receives  the  invoice, 
amounting  to  $29215,  and  on  the  same  date  full  settlement  is  made 
What  amount  is  required  to  cancel  the  bill  ?  (Exact  days,  360 
days  to  the  year.) 

35.  Divide  $2000  in  such  a  manner  between  two  brothers,  aged 
16  and  19  years  respectively,  so  that  when  they  arrive  at  21  years 
of  age  they  will  have  equal  amounts,  money  being  worth  6$  sim- 
ple interest. 

36.  What  would  be  the  share  of  each  if  money  is  worth  6$ 
compound  interest  ? 


314  REVIEW     EXAMPLES.  [Art.  663. 

37.  Find  the  interest  on  $5000  from  May  18  to  Sept.  28,  at  4^: 
1,  Ordinary  interest  and  compound  subtraction ;  2,  Ordinary  in- 
terest and  exact  days  ;  3,  Accurate  interest. 

38.  Find  the  amount  due  on  the  following  note  Jan.  1,  1883, 
by  the  United  States  and  the  Mercantile  Rules  : 


DAVENPORT,  IOWA,  May  1,  1878. 


On  demand,  I  promise  to  pay  EDWIN  D.  MORGAN,  or  order, 
Five  thousand  dollars,  with  interest  at  six  per  cent.,  for  value 
received.  E.  H.  CONGER. 

On  this  note  the  following  payments  were  indorsed  : 
Received  Jan.  16,  1879,  $400.      Received  Dec.  12,  1880,  $150. 
Received  Sept.  7,  1879,  $100.      Received  Aug.  18,  1881,  $850. 
Received  May    1,  1880,  $500.      Received  Apr.  23,  1882,  $100. 

39.  How  much  would  have  been  due  on  the  above  note  at  10$  ? 

40.  What  is  the  value  of  a  draft  on  Hamburg  of  17468  marks 
at  95f  ? 

41.  C.  of  London  owes  me  for  goods   sold   on   my   account, 
£129  18s.  Id.     How  much  do  I  receive  in  payment,  if  I  draw  a 
bill  of  exchange  for  the  amount  and  sell  it  at  4.85-|  ? 

4%.  My  agent  in  Paris  buys  an  invoice  of  merchandise  amount- 
ing to  12488  francs,  at  a  commission  of  2J%.  What  is  the  cost  of 
the  draft  which  I  remit  in  payment,  exchange  being  5.17-f  ? 

43.  An  exporter  sold  the  following  bills  of  exchange  through 
a  broker  :  10000  francs  on  Paris  at  5.16J,  £375  16s.  Sd.  on  Lon- 
don at  4.83-|,  16480  marks  on  Hamburg  at  94  J,  5287  guilders  on 
Amsterdam  at  41  -J.  What  were  the  proceeds,  brokerage  \%  ? 

44'  A  commission  merchant  at  New  York  sells  goods  for  A.  of 
Havre  to  the  amount  of  $3435.27,  and  charges  a  commission  of 
2  J%  for  selling.  What  is  the  face  of  the  draft  which  he  purchases 
and  remits  in  settlement,  exchange  being  5.27  ? 

45.  My  agent  in  London  has  purchased  for  me,  at  a  commis- 
sion of  %\%9  375  dozen  kid  gloves  at  49^.  per  dozen,  and  636  yards 
silk  at  9s.  6d.  per  yard.     When  exchange  is  $4.86|,  what  will  be 
the  cost  of  the  draft  which  J.  remit  to  him  in  settlement  ? 

46.  Purchased  in  England,  merchandise  amounting  to  £324 
10s.  7d.,  and  paid  freight  and  duties  $487.34.     How  much  per  £ 
must  I  sell  these  goods  to  gain  ¥&\%  on  the  full  cost,  and  what 
must  I  charge  for  an  article  invoiced  at  6s.  8d.,  exchange  4.88  ? 

47.  What  is  the  cost  of  insuring  $18000  at  75^  less  15%  ? 


Art.  663.] 


REVIEW     EXAMPLES. 


315 


48.  Average  the  following  account,  and  find  the  amount  due 
Sept.  28,  1882,  at  6$. 

$874.32  on  30  days       credit. 
518.65   "  60     " 
373.78   "    4  months     " 
429.31    "  60  days 
657.70 
242.28 


Mar.  16,  1882, 

"    31,  " 

May     5,  " 

"      21,  " 

June  18,  " 

July     3,  " 

"      24,  " 

Aug.  19,  " 

Sept.  13,  " 


"  30 
"  60 


983.75 
716.30 


4  months 
4       " 


49.  Average  the  following  account, 
due  Jan.  1,  1883  ? 


536.60  "  60  days          " 

What  will  be  the  amount 


Dr. 


DANIEL  S.  LAMONT,  Albany,  N.  Y. 


Cr. 


1882, 

1882. 

i 

July  16 

Mdse.,  4  mo. 

$876 

14 

;  Sept.  10 

Cash,  .  .  . 

$900 

00 

Aug.  4 

"  60  da. 

415 

65 

"  21 

tt 

700 

00 

Sept.  10 

"  30  da. 

797 

38 

Oct.  13 

(C 

500 

00 

"  21 

"  30  da. 

686 

96 

"  31 

Mdse.,  30  da. 

322 

16 

Oct.  13 

"   4  mo. 

524 

27 

Nov.  2 

Cash,  .  .  . 

400 

00 

"  31 

"  30  da. 

859 

75 

"  28 

Note,  4  mo.  . 

800 

00 

Nov.  28 

"  60  da. 

263 

31 

Dec.  27 

Cash,  .  .  . 

500 

00 

Dec.  1 

"  60  da. 

172 

64 

"  30 

"  30  da. 

938 

52 

50.  Prepare  an  account  current,  including  interest  at  Q%  to 
Jan.  1,  1883,  from  the  above  ledger  account,  according  to  the  form 
and  method  of  Art.  59O. 

51.  Sold  five  $1000  bonds  at  116f,  and  invested  the  proceeds 
in  railroad  stock  at  92$,  which  I  sold  at  98£.     What  was  the  gain 
on  the  stock,  allowing  usual  brokerage  ? 

62.  Sold  Aug.  11,  1879,  500  shares  Chicago  &  Alton,  at  94J, 
and  covered  my  short  sale  Aug.  16,  1879,  at  91.  What  was  my 
profit,  allowing  the  usual  brokerage  ? 

53.  What  annual  income  will  be  obtained  by  investing  $9923. 75 
in  bonds,  bearing  5%  interest,  and  purchased  at  116f  ? 

54.  Bought  stock  at  116f  and  sold  at   112f     Loss,   $1295. 
What  was  the  par  value  of  the  stock  ? 


316  REVIEW     EXAMPLES.  [Art.  663. 

55.  The  tax  levied  in  a  town,  having  a  valuation  of  $1800000, 
is  $22500.     What  is  the  tax  on  $1,  and  what  is  the  tax  of  A, 
whose  real  estate  is  assessed  $5000  and  personal  property  $1500  ? 

56.  What  is  the  duty  at  60%  on  an  invoice  of  silk  amounting  to 
36475  francs  ? 

57.  What  is  the  duty  on  50  cwt.  3  qr.  14  Ib.  (Long  ton  table) 
of  steel  at  2Jc\  per  pound  ? 

58.  Find  the  duty  at  25%  on  an  invoice  of  mdse.  valued  at 
£243  2s.  3d. 

59.  What  is  the  duty  on  a  block  of  marble  2  x  3  x  7  ft.,  im- 
ported from  Italy,  dutiable  value  3450  lire,  and  duty  $1  per  cubic 
foot  and  25%  ? 

60.  A,  B,  and  C  are  partners  in  business,  investing  as  follows  : 
A,  $4000 ;  B,  $6000 ;  C,  $8000.     The  partners  are  to  share  the 
profits  and  losses  in  proportion  to  their  investments.     Each  is 
entitled  to  compensation  for  services  at  the  rate  of  $150  per  month, 
to  be  credited  the  first  day  of  the  following  month.     Interest  is  to 
be  reckoned  on  the  salaries  and  on  the  amounts  drawn  out  at  the 
rate  of  6%  per  annum.     At  the  end  of  the  year  B  and  C  purchase 
the  interest  of  A,  and  in  the  payment  therefor,  it  is  desired  that 
the  remaining  members  shall  so  invest  that  their  interests  shall 
be  equal.     It  is  mutually  agreed  that  the  "good  will"  of  the 
business  shall  be  valued  at  $3000  in  the  final  settlement.     It  is 
also  agreed  that  a  discount  of  5%r?hall  be  allowed  upon  all  uncol- 
lected  accounts  as  a  fund  to  meet      i  debts  and  costs  for  collecting. 
A  statement  of  the  business  previous  to  closing  shows  the  follow- 
ing results :  merchandise,  horses,  wagon,  office  fixtures,  and  cash 
on  hand,  $12410;  sundry  debtors,  $17030;  sundry  creditors,  $4050; 
expense  account  (not  including  partners'  salaries),  $2400 ;  profit 
on  merchandise  sold,  $15290.     A  withdrew  on  account  of  salary 
Apr.  1,  $450  ;  July  1,  $300 ;  Oct.  1,  $400.     B  withdrew  Mar.  1, 
$400 ;  Apr.  1,  $150  ;  June  1,  $400  ;  Oct.  1,  $800  ;   Dec.  1,  $500. 
C  withdrew  Apr.  1,  $600 ;  July  1,  $700 ;  Oct.  1,  $600 ;  Nov.  1, 
$200.     How  much  must  B  and  0  each  invest  or  pay  A,  and  how 
should  the  books  of  the  new  firm  be  opened  ? 

NOTES. — 1.  B  and  C,  not  desiring  to  have  the  new  books  encumbered  with 
the  contingent  accounts  of  "good  will"  and  "reserve  fund,"  closed  these 
accounts  after  a  settlement  was  made  with  A. 

2.  The  loss  or  gain  may  be  found  from  a  statement  of  resources  and 
liabilities,  or  from  the  Loss  and  Gain  account. 


APPENDIX. 


GREATEST    COMMON    DIVISOR. 

664.  The   Greatest   Common  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  divide  each  without  a 
remainder ;  hence  it  is  their  greatest  common  factor. 

Thus,  2,  3,  4,  and  12  are  common  divisors  of  36,  48,  and  60;  12  is  their 
greatest  common  divisor. 

665.  To  find  the  greatest  common  divisor  of  two  or 
more  numbers. 

Ex.  What  is  the  greatest  common  divisor  of  168,  252,  and  420  ? 

OPERATION.  ANALYSIS. — Divide  the  given  numbers  by 

4  )  168,   252,   420  any  number  that  will  divide  them  all  without 

IJF  \  ^2       j^3 Jog  a  remainder,  and  divide  the  quotients  in  the 

same  manner  until  the  last  quotients  have  no 

3  )  6,        9,      IP  common  divisor.     Since  4  will  divide  all  the 

235  given   numbers,  and    3    and    7   will    divide 

SUCCP^  vely   the    resulting  quotients,    their 

4  X  7  X  3  =  84.  prod     [,  84,  is  a  common  divisor  of  the  given 

numbers.     Since  the  last  quotients  have  no 
common  divisor  or  factor,  84  is  the  greatest  common  divisor. 

666.  RULE. — Divide  the  given  numbers  by  any  factor 
that  will  divide  all  of  them  without  a  remainder.    In 
like  manner  divide  the  resulting  quotients,  and  continue 
the  division  until  the  quotients  have  no  common  factor. 
The  product  of  the  several  divisors  will  be  the  greatest  com- 
mon divisor. 

EXAMPLES. 

667.  Find   the   greatest   common   divisor   of   the   following 
numbers : 

1.  108,  144,  and  360.  5.  405,  243,  and  324. 

0.  144,  336,  and  240.  6.  378,  126,  and  252. 

3.  165,  550,  and  220.  7.  375,  625,  and  250. 

4.  792,  144,  and  216.  8.  288,  720,  and  864. 


318  GREATEST     COMMON    DIVISOR.          [Art.  668. 

668.  To  find  the  greatest  common  divisor   of  two 
numbers  when  they  are  not  readily  factored. 

669.  PRINCIPLES.  —  1.  A  common  divisor  of  two  numbers  is 
a  divisor  of  their  sum,  and  also  of  their  difference. 

2.  A  divisor  of  a  number  is  a  divisor  of  any  multiple  of  that 
number. 

670.  RULE.  —  Divide  the  greater  number  by  the  smaller, 
and  divide  the  last  divisor  by  the  remainder  ;    and   so  con- 
tinue until  there  is  no  remainder.     The  last  divisor  will  be 
the  greatest  common  divisor. 

NOTES.  —  1.  When  the  greatest  common  divisor  of  more  than  two  numbers 
is  required,  find  the  greatest  common  divisor  of  the  smallest  two  first,  and  of 
this  greatest  common  divisor  and  the  next  greater,  and  so  on,  until  all  the 
numbers  are  used.  The  last  divisor  will  be  the  greatest  common  divisor  of 
all  the  given  numbers. 

3.  If  the  remainder  at  any  time  is  a  prime  number,  and  it  is  not  contained 
in  the  last  divisor,  there  is  no  common  divisor  greater  than  1  ;  it  will  there- 
fore be  useless  to  further  continue  the  division. 

Ex.  Find  the  greatest  common  divisor  of  391  and  437. 

OPERATION.  DEMONSTRATION.—  Since  23  is  a  divisor  of  46,  it 

391  )  437  (1  is  a  divisor  of  368,  a  multiple  of  46  (Prin.   2). 

39^  Since  23  is  a  divisor  of  itself  and  368,  it  is  a  divisor 

-  of  their  sum,  391  (Prin.  1).     Since  23  is  a  divisor 

46  )  391  (8  Of  46  ana  391,  it  is  a  divisor  of  their  sum,  437.     23 

368  is  therefore  a  common  divisor  of  391  and  437,  the 

~~  numbers. 


23)46(2 

.„  The  greatest  common  divisor  of  391  and  437, 

_          whatever  it  may  be,  is  a  divisor  of  their  difference, 

0  46  (Prin.  1)  ;  also  of  368,  a  multiple  of  46  (Prin.  2)  ; 

also  of  23,  391  —  368  (Prin.  1).     Since  the  divisor 

of  a  number  cannot  be  greater  than  itself,  the  greatest  common  divisor  of  the 
given  numbers  cannot  be  greater  than  23.  23  is  therefore  the  greatest  common 
divisor. 

EXAMPLES. 

671.   Find  the   greatest   common  divisor   of    the   following 
numbers  : 

1.  319  and  377.  &  611,  799,  and  987. 

&  259  and  629.  6.  744,  984,  and  522. 

3.  589  and  713.  7.  391,  667,  and  920. 

4.  903  and  989.  8.  451,  481,  and  737. 


Ait.  672.]  ANNUAL     INTEREST.  319 


ANNUAL     INTEREST. 

672.  When  a  note  contains  the  words  "with  interest  annu- 
ally," the  laws  of  New  Hampshire  and  Vermont,  if  the  interest 
is  not  paid  when  due,  allow  simple  interest  on  the  annual  interests 
from  the  time  they  become  due  to  the  time  of  payment. 

ILLUSTRATION. — A  agrees  to  pay  B  $6000  in  three  years  from  Jan.  1, 
1880,  with  interest  annually  at  6%.  By  this  contract,  $360  becomes  due 
Jan.  1,  1881,  and  on  the  first  day  of  January  in  each  year  thereafter,  until 
paid  ;  this  is  the  "  annual  interest."  Suppose  A  does  not  pay  any  portion  of 
this  interest  until  Jan.  1,  1883,  when  the  principal  becomes  due  ;  then  A,  hav- 
ing had  the  use  of  money  that  his  contract  required  him  to  pay  to  B,  and  B 
having  been  deprived  of  its  use,  B  is  entitled  to  have  simple  interest  added  to 
the  annual  interest,  from  the  time  when  the  same  became  due  to  Jan.  1, 1883 ; 
BO  that  on  Jan.  1,  1883,  B  would  be  entitled  to  the  following  sums  as  interest : 


First  year's  int.  $360  +  2  yrs.  simple  int.  thereon,  $43.20  = 
Second"        "      360  +  1    "        "         "          "          21.60=      381.60 
Third    "        "      360  +  0  (paid  when  due)  00  =      360 

$1080  $64.80  =  $1144.80 

Amount  of  annual  interest $1080.00 

Amount  of  simple  interest  accrued  upon  annual  interest    .  64.80 

Total  amount  of  interest  due $1144.80 

In  calculating  the  simple  int.  upon  the  annual  int.,  shorten  the  operation 
by  finding  the  int.  upon  the  annual  int.  for  the  sum  of  the  several  periods. 

Ex.  What  is  the  amount  due  on  the  following  note  July  1, 
1885? 

$10000.  CON-CORD,  1ST.  H.,  January  1,  1882. 

Three  years  after  date,  for  value  received,  I  promise  to  pay 
A.  B.  THOMPSON,  or  order,  Ten  Thousand  Dollars,  with  interest 
payable  annually. 

C.  A.  DOWNS. 

OPEKATION. 

Face  of  note,  on  interest  from  Jan.  1,  1882 $10000.00 

Interest  from  Jan.  1,  1882,  to  July  1, 1885,  3  yr.  6  mo 2100.00 

3  items  of  annual  interest  ($600  each)  are  unpaid : 
1st  from  Jan.  1,  1883,  to  July  1, 1885,        2  yr.  6  mo. 
2nd  from  Jan.  1, 1884,  to  July  1, 1885,        1  yr.  6  mo. 
3rd  from  Jan.  1,  1885,  to  July  1,  1885,  6  mo. 

Int.  on  the  annual  int.  =  int.  on  $600  for  4  yr.  6  mo 162.00 

Total  amount  due  July  1,  1885 $12262.00 


320  APPENDIX.  [Art,  673. 

673.  EULE. — To  the  given  principal  and  its  interest  to 
the  date  of  settlement,  add  the  interest  on  each  annual 
interest  from  the  time  it  is  due  to  the  date  of  settlement. 
Tlic  sum  will  be  the  amount  due  at  annual  interest. 

EXAMPLES. 

674.  1.  At  6%,  interest  payable  annually,  how  much  would 
be  due  Oct.  1,  1884,  according  to  the  laws  of  New  Hampshire,  on 
a  note  of  $8000,  dated  June  1,  1881,  no  payments  having  been 
made? 

2.  What  amount  would  be  due  Jan.  1,  1886,  at  6$,  on  a  note 
for  $4200,  dated  Concord,  N.  H.,  May  16,  1882,  interest  payable 
annually,  and  no  payments  having  been  made  ? 

3.  A  note  for  $10000  was  dated  Apr.  1,  1882,  and  payable 
four  years  from  date  without  interest.    Attached  to  this  note 
were  8  notes   of  $400  each  for  the   semi-annual    interest  due 
Oct.  1,  1882,  Apr.  1,  1883,  Oct.   1,  1883,  Apr.  1,  1884,  Oct.  1, 
1884,  Apr.  1,  1885,  Oct.  1,  1885,  Apr.  1,  1886.     How  much  was 
due,  at  8$,  Apr.  1,  1886,  nothing  having  been  paid  ? 

NOTE. — It  is  the  custom  of  certain  corporations  when  making  loans  for 
long  periods  of  time  on  collateral  security  or  on  bond  and  mortgage,  to  have 
a  note  or  mortgage  given  without  interest  for  the  principal,  and  to  have 
separate  notes  given  for  each  sum  of  annual,  semi-annual,  or  quarterly 
interest,  due  and  maturing  at  the  time  the  interest  is  payable.  These  notes 
draw  interest  after  maturity  like  any  other  note,  and  may  be  collected  without 
disturbing  the  original  loan. 

4.  What  amount  would  be  due  July  1,  1884,  on  a  note  of 
$5000,  dated  July  1,  1882,  given  for  2  years,  with  notes  for  quar- 
terly interest,  no  payments  having  been  made  ? 

5.  Kequired  the  amount  due  Jan.  1,  1883,  on  a  note  of  $3600, 
dated   Jan.   1,  1881,   due   in  two   years,   notes  for   semi-annual 
interest  from  date,  at  6%,  having  been  given,  and  nothing  having 
been  paid. 

6.  Find  the  amount  of  $1200,  at  6^,  interest  payable  annually, 
from  June   16,    1882,   to    Dec.    28,    1886,   no    interest    having 
been  paid  except  for  the  first  year. 

7.  What  must  be  paid,  Oct.  16,  1885,  in  settlement  of  a  note 
for  $2500,  dated  Manchester,  1ST.  H.,  May  6, 1880,  said  note  promis- 
ing interest  annually,  and  no  interest  having  been  paid  ? 


Art.  675.]  NX  W    HAMPSHIRE     RULE.  321 

NE\V     HAMPSHIRE     RULE.* 

675.  According  to  the  laws  of  New  Hampshire,  when  pay- 
ments are  made  upon  a  note,  or  other  contract,  by  virtue   of 
which  interest  is  payable  annually  (672),  they  should  be  applied 
in  the  following  order  to  the  payment  of — 

1.  Any   simple   interest  that   may   have   accrued    upon    the 
annual  interest. 

2.  The  annual  interest.  3.  The  principal. 

676.  EULE. — Find  the  interest  due  upon  the  principal 
and  the  annual  interest  at  the  annual  rest  (the  time  when 
the  annual  interest  becomes  due  from  year  to  year)  next 
after  the  first  payment.      To   the  payment  or  payments 
made  before  this  rest,  add  interest  from  the  dates  when 
they  ivere  made  to  the  date  of  the  rest,  unless  there  is  no 
interest  due  upon  the  principal,  excepting  that  which  is 
accruing  during  the  year  in  which  the  payment  or  pay- 
ments were  made,  and  the  payments  together  are  less  than 
the  interest  thus  accruing,  in  which  last  case  no  interest  is 
to  be  added   to  the  payments.      Deduct  the  payment  or 
payments,  with  or  without  interest,   as    aforesaid,   from 
the    amount    of   principal,   annual    interest,   and  simple 
interest  upon    the    annual  interest   due    at   the   time  of 
said  rest,  if  such  payment  or  payments  equal  or  exceed 
the  annual  and  simple  interest  then  due  ;  if  less  than  such 
annual  and  simple  interest,  but  greater  than  the  simple 
interest  due  upon  the  annual  interest,  deduct  the  same 
from  the  sum  of  the  annual  and  simple  interest,  and  upon 
the  balance  of  such  annual  interest  find  simple  interest  to 
the  time  when  the  next  payment  or  payments  are  applied  ; 
if  less  than  the  simple  interest  due    upon    the    annual 
interest,  deduct  the  same  from  such  simple  interest  and 
add  the  balance  without  interest  to  the  other  interest  due 
at  the  time  when  the  next  payment    or    payments    are 
applied. 

Proceed  in  like  manner  to  the  time  of  the  first  annual 
rest  following  the  next  payment,  and  to  the  end  of  the  time 
required. 

*  From  Report  of  State  Superintendent  of  Public  Instruction  (1877). 


322 


APPENDIX. 


[Art.  677. 


EXAM  PLES. 


677.  1.  According  to  the  law  of  New  Hampshire,  how  much 
is  due  Jan.  1,  1886,  on. a  note  dated  Jan.  1,  1880,  for  $2000,  with 
interest  annually  at  6%,  the  following  payments  having  been 
made :  July  1,  1882,  $500 ;  Oct.  1,  1883,  $50. 


15 


OPERATION. 

First  annual  interest  due  Jan.  1,  1881,  $120  +  2  yr.  simple  interest 
thereon,  $14.40 

Second  annual  interest  due  Jan.  1, 1882,  $120  + 1  yr.  simple  interest 
thereon,  $7.20 • 

Third  annual  interest  due  Jan.  1,  1883, 

Principal 

First  payment,  July  1,  1882,      .... 
Interest  thereon  from  July  1, 1882,  to  Jan.  1,  1883, 

Balance  of  principal  due  Jan.  1,  1883, 

Fourth  annual  interest  of  $1866.60,  due  Jan.  1,  1884, 
Second  payment,  Oct.  1,  1883  (being  less  than  the  interest  accruing 
during  the  year,  it  does  not  draw  interest)    .        . 

Balance  of  fourth  annual  interest  unpaid 

Fifth  annual  interest  of  $1866.60,  due  Jan.  1,  1885, 

Sixth  annual  interest  of  $1866.60,  due  Jan.  1,  1886, 

Simple  interest  on  unpaid  balance  of  fourth  annual  in.*,  for  %  yr.   . 

Simple  interest  on  fifth  annual  interest  for  1  year   . 

Balance  of  principal 

Amount  due  Jan.  1, 


$134.40 

127.20 

120.00 

2000.00 

$2381.60 

515.0ft 

1866.60 

112.00 

50.00 


62.00 
112 
112 
7.44 
6.72 
1866.60 
2166.76 

Solve  Examples  2,  4,  8,  and  9,  Art.  5O5,  according  to  the  New 
Hampshire  Rule,  at  the  legal  rate  (436),  supposing  each  note  to 
contain  the  words  "with  interest  annually." 

VERMONT     RULE. 

678.  The  Vermont  Rule  for  notes  with  interest  is  essentially 
the  same  as  the  United  States  Rule  (5O4) ;  and  for  notes  "with 
interest  annually,"  it  is  the  same  as  the  New  Hampshire  Rule, 
except  that  when  payments  are  made  on  account  of  interest  accru- 
ing but  not  yet  due,  they  draw  interest  from  the  date  they  were 
made  to  the  annual  rest,  whether  they  are  greater  or  not  than  the 
interest  accruing  during  the  year. 

Thus,  by  the  Vermont  Rule,  the  payment  of  $50,  in  the  above  example, 
would  draw  interest  from  Oct.  1,  1883  to  Jan.  1,  1884,  or  3  months.  The 
unpaid  balance  of  fourth  annual  interest  would  be  $61.25  ($112  —  $50.75). 


STORAGE. 


679.  Storage  is  keeping  or  storing  of  goods  in  a  warehouse 
until  they  are  required  for  use,  sale,  or  transportation. 

Storage  is  also  the  name  applied  to  the  price  or  compensation  for  storing1 
goods  in  a  warehouse. 

680.  Storage  is  usually  calculated  at  a  certain  rate  per  barrel, 
bale,  bushel,  box,  or  other  unit  for  a  certain  time. 

1.  The  storage  term  is  one  week,  10  days,  20  days,  or  one  month. 

2.  In  some  warehouses,  storage  for  a  part  of  a  term  is  charged  at  the  same 
rate  as  for  a  full  term. 


(j  CASH    STORAGE. 

681.  When  the  storage  is  paid  or  estimated  when  the  goods 
are  taken  out  of  store  or  the  receipt  is  surrendered,  it  is  sometimes 
called  cask  storage. 

EXAMPLES. 

682.  1.  What  was  paid  for  the  following  storage  at  6  cents 
per  barrel  per  month  or  part  of  a  month,  the  calculation  being 
made  at  each  delivery  ?     Received  Oct.  1,  1800  III. ;  Nov.  15,  360 
lll.\   Dec.   18,  420  III.-,  Dec.  27,  432  III.      Delivered  Oct.  31, 
1000  III.-,  Dec.  4,  240  ML;   Dec.  19,   600  III.-,  Dec.  26,  300  III. 


OPEKATION. 


Date. 

Received. 

Delivered. 

Oct.      1 

1$00 

"      31 

Nov.  15 

"5^19, 
$00 

1000 

Dec.     4 
."     18 

2  0 

420 

000 

"     19 

$00 

"     26 

On  hand 

432 

872 

1000 

240 

560 

40 

300 


.06  .== 
.18'  = 


60.00 
43.20 


,18  —  100.80 


12  = 

12  = 


4.80 
36.00 

$244.80 


ANALYSIS. — All  goods  delivered  are 
deducted  from  the  oldest  receipt  on 
hand.  By  the  system  of  cancellation 
indicated  in  the  operation,  it  can  be 
easily  determined  when  the  storage 
commences.  The  1000  bbl.  taken  out 


324 


APPENDIX. 


[Art.  682. 


Oct.  31  were  placed  in  store  Oct.  1,  and  pay  1  month's  storage.  The  240  bbl. 
taken  out  Dec.  4  were  placed  in  store  Oct.  1,  and  pay  3  months'  storage.  Of 
the  delivery  of  600  bbl.,  560  were  placed  in  store  Oct.  1,  and  pay  3  months' 
storage,  and  the  remainder,  40  bbl.,  were  placed  in  store  Nov.  15,  and  pay  2 
months'  storage,  Dec.  19.  The  lot  of  300  bbl.  withdrawn  Dec.  26,  were  placed 
in  store  Nov.  15,  and  pay  2  months'  storage.  The  separate  calculations  are 
placed  at  the  right  in  the  above  operation. 

NOTES. — 1.  Certain  warehouses  render  bills  at  the  end  of  each  month  for 
all  goods  taken  out  during  the  month.  Others  render  bills  monthly  for  all 
storage  dues,  whether  the  goods  have  been  withdrawn  or  not. 

2.  Storage  on  goods  for  which  negotiable  receipts  have  been  issued,  and  in 
many  other  cases,  is  collected  when  the  receipt  is  surrendered  or  the  goods 
delivered. 

2.  What  will  be  the  storage  at  5  cents  per  barrel  per  month 
on  the  following  ?     Received  Aug.  1,  800  bbl. ;  Aug.  15,  700  bbl. ; 
Aug.  26,  900  bbl.     Delivered  Aug.  12,  400  bbl;  Aug.  20,  800  bbl ; 
Sept.  1,  400  bbl ;  Sept.  8,  800  bbl 

3.  Find  the  storage  due  on  the  following  account  June  24,  at 
3  cents  a  bale  per  month  or  part  of  a  month.     Received  Apr.  13, 
400  bales  ;  Apr.  30,  800  bales ;  May  16,  200  bales ;  May  25,  400 
bales  ;  June  19,  600  bales.     Delivered  May  10,  600  bales  ;  May  20, 
100  bales  ;  May  28,  700  bales ;  May  31,  400  bales ;  June  24,  600  bales. 

4.  Complete  the  following  storage  bill,  the  rate  being  lOc.  per 
bale  per  month  or  part  of  a  month. 


Messrs.  ARMSTRONG,  CATOR  &  Co., 


BALTIMORE,  MD.,  Aug.  31,  1887. 
To  MERCHANTS'  STORAGE  Co.,  Dr. 


Marks  and  Numbers. 

When 
received. 

When 
delivered. 

Rate. 

Amount. 

4 

Bales  AC  $174-177 

June   4 

Aug.  2 

20c. 

80 

1 

A  C$298 

May  21 

4 

30c. 

30 

5 

A.  C.  &  Co.  121-25 

July  26 

7 

lOe. 

50 

3 

AC  $170-1  72 

June  4 

16 

** 

** 

2 

C  $  29-28 

Aug.  17 

20 

** 

** 

6 

A.  C.  &  0^115-20 

July  26 

22 

*# 

* 

*# 

1 

A  C$173 

June  4 

24 

** 

#* 

4 

AC  it  299-302 

May  21 

27 

** 

* 

** 

5 

A.  C.  &  Co.  $26-30 

July  26 

31 

•H-* 

* 

* 

** 

NOTE. — In  many  cases  (see  above  example),  storage  is  charged  for  the 
time  the  particular  packages  withdrawn  have  been  in  store. 

5.  The  following  quantities   of   wheat   were   stored  at   Ic.  a 
bushel  per  month  or  part  of  a  month.    What  was  the  amount  of 


Arc.  682.] 


STOBA  G  E. 


325 


storage  due  June  1,  a  full  settlement  being  made  on  that  date  and 
a  new  receipt  being  given  ?  Received  Apr.  1,  600  bushels  ;  Apr. 
25,  400  bushels ;  May  9,  400  bushels ;  May  27,  300  bushels. 


AVERAGE     STORAGE. 

683.  At  some  warehouses,  in  computing  storage  on  grain, 
flour,  etc.,  when  there  are  frequent  receipts  and  deliveries,  it  is 
customary   to  average  the  time  and  charge  a  certain   rate   per 
month  of  30  days.      The  process  is  called  average  storage,  or 
storage  on  account. 

684.  Ex.   Merchandise  was  received  and  delivered  at  a  ware- 
house as  follows :   Received  Oct.  1,  1800  III.  flour ;  Nov.  15,  360 
bbl;  Dec.  18,  420  bbl;   Dec.    27,  432  III.     Delivered  Oct.   31, 
1000  bbl;  Dec.  4,  240  III.-,  Dec.  19,  600  bbl-9  Dec.   26,  300  bbl 
Find  the  average  storage  due  on  the  above  Jan.  1,  at  6  cents  per 
barrel  per  month  of  30  days. 


685.  OPERATION. — PRODUCT  METHOD. 


Date. 

Received. 

Days. 

Products. 

Date. 

Delivered. 

Days. 

Products. 

Oct.         1 

1800 

92 

165600 

Oct.  31 

1000 

62 

62000 

Nov.     15 

360 

47 

16920 

Dec.    4 

240 

28 

6720 

Dec.     18 

420 

14 

5880 

"      19 

600 

13 

7800 

27 

432 

5 

2160 

"      26 

300 

6 

1800 

3012 

190560 

2140 

78320 

2140                      78320 

On  hand 


872 


30  )  112240 


3741^  x  .06  =  224.48 


ANALYSIS. — Assuming  that  there  was  nothing  withdrawn,  the  1800  bbl. 
would  be  in  store  from  Oct.  1  to  Jan.  1,  or  92  days.  The  storage  of  1800  bbl.  for 
92  days  is  equivalent  to  the  storage  of  1  bbl.  for  165600  days.  The  storage  of 
360  bbl.  for  47  days  is  equivalent  to  the  storage  of  1  bbl  for  16920  days.  In 
the  same  manner,  we  find  the  total  storage,  if  nothing  had  been  withdrawn, 
to  be  equivalent  to  the  storage  of  1  bbl.  for  190560  days.  The  storage  on 
the  goods  withdrawn  is  equivalent  to  the  storage  of  1  bbl.  for  78320  days, 
thus  making  the  net  storage  I 'bbl.  for  112240  days,  or  3741£  months.  6  cents 
multiplied  by  3741£  equals  $224.48,  the  total  storage. 


326 


APPENDIX. 


[Art.  686. 


686.  OPERATION. — BY  DAILY  BALANCES. 


Date. 

Received. 

Delivered. 

Balances. 

Days. 

Products. 

Oct.     1 

1800 

1800 

30 

54000 

"      31 

1000 

800 

15 

12000 

Nov.  15 

360 

1160 

19 

22040 

Dec.     4 

240 

920 

14 

12880 

"      18 

420 

1340 

1 

1340 

"      19 

600 

740 

ry 
I 

5180 

"      26 

300 

440 

1 

440 

"     27 

432 

872 

5 

4360 

3012 

2140 

92 

30  )  112240 

Jan.      1       Bal.  on  hand      872                                                   3741J 

3012            3012       3744  x  .06  =  224.48 

ANALYSIS. — Arrange  the  receipts  and  deliveries  in  the  order  of  their 
dates  as  in  the  operation.  Find  the  number  of  barrels  on  hand  at  each  of  the 
dates.  The  1800  bbl.  are  in  store  from  Oct.  1  to  Oct.  31,  or  30  days.  The 
storage  of  1800  bbl.  for  30  days  is  equivalent  to  1  bbl.  for  54000  days.  The 
total  storage  is  equivalent  to  the  storage  of  1  bbl.  for  112240  days,  or  3741£ 
months.  6  cents  multiplied  by  3741^  equals  $224.48,  the  total  storage. 

EXAMPLES. 

687.  1.  Find  by  either  of  the  above  methods  the  average 
storage  at  5  cents  per  month  of  30  days,  of  the  account  given  in 
Ex.  2,  Art.  682. 

2.  Find  the  average  storage,  at  3  cents  per  month  of  30  days, 
of  the  account  given  in  Ex.  3,  Art.  682. 

3.  Find  the  total  charge  for  pasturing  cattle  per  the  following 
statement,  at  30  cents  a  head  per  week  :    Received  July  5,  18 
head ;  July  12,  10  head  ;  July  20,  30  head ;  Aug.  2,  40  head'; 
Aug.  10,  20  head ;  Sept.  1,  10  head ;  Sept.  4,  24  head  ;  Sept.  17, 
26  head;  Oct.  2,  20  head  ;  Oct.    27,  18  head;  Nov.    1,  6  head; 
Nov.  2,  16  head.     Withdrawn  July  7,  4  head;  July  9,  8  head; 
July  14,  10  head ;  July  17,  6  head  ;  July  23,  20  head  ;  Aug.  4, 
20  head  ;  Aug  13,  20  head  ;  Aug.  21,  12  head  ;  Aug.  29,  4  head  ; 
Sept.   8,  10  head ;  Sept.  14,  30  head ;  Sept.   21,  18  head  ;  Oct. 
30,  20  head ;  Nov.  5,  10  head ;  Nov.   9,  20  head ;  Nov.  16,  26. 
head. 

NOTE. — When  the  account  is  long,  the  first  method  is  preferable. 


ALLIGATION. 


688.  Alligation  treats  of  mixing  ingredients  of  different 
values  to  find  the  value  of  the  mixture,  or  to  produce  a  mixture 
of  a  given  value. 

NOTE. — Alligation  is  sometimes  and  more  properly  called  Average. 

ALLIGATION     MEDIAL. 

689.  Alligation  Medial  is  the  process  of  finding  the  average 
value  of  a  mixture,  the   rates  and   quantities  of  the  ingredients 
being  given. 

EXAMPLES. 

690.  1.  A  grocer  mixes  together  7  pounds  of  coffee  at  26 
cents  per  pound,  4  pounds  at  27  cents  per  pound,  and  10  pounds 
at  34  cents  per  pound.     What  is  the  value  of  a  pound  of  the 
mixture  ? 

OPERATION. 

7  x  26c.    =  $1.82  ANALYSIS.— 7  Ib.  at  26c.  are  worth  $1.82.   4  Ib. 

4  x  27 C.  =  1.08  at  27c.  are  worth  $1.08.  10  Ib.  at  34c.  are  worth 
10  X  34c.  —  3.40  $3.40.  Hence  the  total  mixture  containing  21  Ib. 
is  worth  $6.30,  and  1  Ib.  is  worth  $6.30  •*-  21,  or 
21  Ib.  worth  $6.30  30c> 

1  II.      "  .30 

2.  A  wine  merchant  mixed  together  10  gallons  of  wine  at  40 
cents  a  gallon,  15  gallons  at  50  cents,  and  25  gallons  at  70  cents. 
What  is  the  value  of  a  gallon  of  the  mixture  ? 

8.  A  grocer  mixed  60  Ib.  of  tea  at  25  cents  a  Ib.,  75  Ib.  at  30 
cents,  and  65  Ib.  at  50  cents.  What  was  the  value  of  a  pound  of 
the  mixture  ? 

4.  A  farmer  mixes  together  20  lu.  of  oats  at  40  cents  a  bushel, 
30  bu.  of  corn  at  50  cents  a  bushel,  and  50  bu.  of  rye  at  70  cents 
a  bushel.  What  is  the  value  of  a  bushel  of  the  mixture  ? 


APPENDIX. 


[Art.  691. 


ALLIGATION     ALTERNATE. 

691.  Alligation   Alternate  is  the  process  of  finding  the 
quantities  of  different  values  required  to  produce  a  mixture  of  a 
given  value. 

692.  The  values  of  several  ingredients  being  given, 
to  produce  a  mixture  of  a  given  value. 

EXAMPLES. 

693.  1.  How  much  tea   worth  24,  28,  36,  and  42  cents  a 
pound  must  be  mixed  together,  so  that  the  mixture  will  be  worth 
30  cents  a  pound  ? 


OPERATIONS 


30 


(1)   (2}   (3)   (4)    (5)  (6}  (7)  (8)  (9) 


24 

* 

0 

i 

0 

1 

9 

/v 

3  |  i  ;  3 

etc. 

28 

0 

} 

0 

6 

6 

6 

6  ]  12    12 

etc. 

36 

i 

0 

1 

0 

1 

2 

3  I  1  ;  3 

etc. 

42 

0 

A 

0 

1 

1 

1 

1      2  |  2 

etc. 

ANALYSIS. — If  we  sell  1  pound  for  30  cents,  that  is  worth  24  cents,  we 
gain  6  cents,  and  to  gain  1  cent,  we  take  |  of  a  pound.  We  must  now  take 
a  kind  that  is  worth  more  than  the  average  price  so  as  to  lose  one  cent.  If  we 
take  a  pound  worth  36  cents  and  sell  it  at  30  cents,  we  will  lose  6  cents,  and  to 
lose  one  cent,  we  must  take  ^  of  a  pound.  In  the  same  manner,  we  find  that 
if  we  take  ^  of  a  pound  of  the  28-cent  tea  and  mix  it  with  ^  of  a  pound  of 
the  42-cent  tea,  there  will  be  no  gain  nor  loss  by  selling  at  30  cents  a  pound. 

I  is  to  £  as  1  is  to  1,  and  |  (^)  is  to  TV  as  6  is  to  1.  Or,  columns  3  and  4 
may  be  found  by  multiplying  columns  1  and  2  respectively  by  the  least  com- 
mon denominators  of  the  fractions. 

Column  5  is  the  sum  of  columns  3  and  4.  An  unlimited  number  of 
answers  may  be  found  to  examples  of  this  kind  by  combining  1,  2,  or  3,  etc., 
times  column  3  with  1,  2,  or  3,  etc.,  times  column  4. 

2.  A  grocer  has  sugar  at  5^,   7^,   12^,  and  13^  per  pound. 
How  much  of  each  kind  will  form  a  mixture   worth   10   cents 
per  pound  ? 

3.  A  jeweler  wishes  to  make  a  compound  of  gold  that  shall  be 
20  carats  fine.     He  has  gold  of  15,  19,  23,  and  24  carats  fine. 
What  quantity  of  each  must  he  take  ? 

4-  How  much  tea  at  25  cents,  50  cents,  60  cents,  and  80  cents 
per  pound  must  be  taken  to  form  a  mixture  worth  55  cents  per 
pound  ? 


Art.  693.J 


ALL1G  A  T10N. 


329 


5.  How  much  wine  at  50  cents,  70  cents,  80  cents,  $1.00,  and 
$1.20  a  gallon  must  be  mixed  together  that  the  mixture  may  be 
worth  90  cents  a  gallon  ? 

694.  When  the  quantity  of  one  ingredient  is  given. 


EXAM  PLES. 


695.  1.  How  much  coffee  at  30,  34,  and  44  cents  per  pound, 
must  be  mixed  with  10  pounds  at  36  cents  a  pound,  to  make  a 
mixture  worth  40  cents  a  pound  ? 


OPERATION. 


(1)     (2)     (3)     (4)    (S)    (6)     (7)     (8) 


30 

TV 

0 

0 

2 

0 

0 

0 

2 

34 

0 

i 

0 

0 

2 

0 

0 

2 

44 

1- 

t 

i 

5 

3 

1 

10 

18 

36 

0 

0 

i 

0 

0 

1 

10 

10 

40- 


ANALYSIS. — We  find  the  relative  quantities  (columns  4,  5,  and  6)  as  in 
Art.  693.  In  order  to  use  10  pounds  of  the  36-cent  coffee,  we  multiply 
column  6  by  10,  producing  column  7.  Column  8  is  found  by  adding  columns 
4,  5,  and  7. 

Other  combinations  may  be  found,  by  multiplying  columns  4  and  5, 
find  adding  the  results  to  column  7. 

2.  How  much  coffee  at  15c.,  17c.,  and  220.  a  pound  must  be 
mixed  with  5  Ib.  at  180.  per  pound  to  make  a  mixture  worth 
20c.  per  pound. 

3.  How  much  gold  of  21  and  23  carats  fine,  must  be  mixed 
with  30  oz.  of  20  carats  fine,  so  that  the  mixture  may  be  22  carats 
fine  ? 

4-  How  much  tea  at  20  cents,  25  cents,  and  45  cents  a  pound, 
must  be  mixed  with  36  Ib.  at  60  cents  a  pound,  so  that  the  mixture 
will  be  worth  40  cents  a  pound  ? 

5.  How  much  wine  at  $1.25  and  $1.75  a  gallon,  must  be  mixed 
with  15  gallons  of  water,  so  that  the  mixture  may  be  worth  $1  a 
gallon  ? 

6.  How  much  tea  at  30  cents,  46  cents,  and  48  cents  a  pound, 
must  be  mixed  with  12  pounds  at  38  cents,  so  that  the  mixture 
may  be  worth  40  cents  a  pound  ? 


330 


APPENDIX. 


[Art.  696. 


2 

0 

2 

} 

0 

3 

3 

\  x   5  = 

3 

2 

5 

J 

5 

5 

10 

)  50  (  5 

696.  When  the   total  quantity  of  the  ingredients  is 
given. 

EXAMPLES. 

697.  -?.  A  grocer  mixed  tea  worth  20,  25,  and  35  cents  a 
pound.     The  mixture  consisted  of  50  pounds,  worth  29  cents  a 
pound.     How  many  pounds  of  each  did  he  take  ? 

(1)    (2)    (3)    (4)    (5) 
i       0 
0       t 

i 


ANALYSIS. — We  find  columns  3  and  4  as  in  Art.  O93.  Column  5  is  the 
sum  of  columns  3  and  4.  The  required  amount  is  50  =  5  times  10,  the  sum  of 
column  5.  Hence  the  quantity  of  each  may  be  found  by  multiplying  each 
number  in  column  5  by  5. 

NOTE.— Many  results  may  be  obtained  for  examples  of  this  kind.  Thus, 
in  the  above  example,  9  times  column  3  plus  once  column  4,  8  times  column  3 
plus  2  times  column  4,  etc.,  would  each  produce  correct  results. 

2.  How  much  wine  worth  50  cents,  60  cents,  90  cents,  and 
$1.20  a  gallon,  must  be  mixed  together  so  as  to  make  a  hogshead 
of  110  gallons  at  80  cents  a  gallon  ? 

3.  A  man  bought  20  barrels  of  flour  for  $120,  paying  $4J,  $5, 
$6|,  and  $7  per  barrel.     How  many  barrels  of  each  did  he  buy  ? 

OPERATION. 

(1)     («)     (*)     (4)     (5)     (6)     (7)     (8)     (9)    (10)  (11)  (IS) 


*J 

! 

0 

1 

0 

1 

1 

0 

2 

2 

0 

3 

3 

5 

0 

1 

0 

8 

0 

8 

6 

0 

6 

4 

0 

4 

6} 

2 

0 

3 

0 

3 

3 

0 

6 

6 

0 

9 

9 

7 

0 

1 

0 

8 

0 

8 

6 

0 

6 

4 

0 

4 

2 

4 

20 

20 

20 

ANALYSIS. — Find  columns  2  and  3  as  in  Art.  693.  Since  6,  the  sum  of 
columns  2  and  3,  is  not  an  exact  divisor  of  20,  the  required  amount,  we  must 
take  a  certain  number  of  times  column  2  and  a  certain  number  of  times 
column  3.  By  trial,  we  find  that  8  times  column  2  plus  once  column  3  equals 
20.  Therefore  multiply  column  2  by  8,  producing  column  4,  and  column  3  by 
1,  producing  column  5.  Column  6  is  the  sum  of  columns  4  and  5.  In  the 
same  manner,  we  find  the  results  given  in  columns  9  and  12. 

4.  A  man  bought  50  animals. for  $50,  paying  for  lambs  $£ 
.each,  for  sheep  $1J  each,  and  for  calves  $3J  each.  How  many  of 
each  did  he  buy  ? 


Art.  698.]  SQUARE     ROOT.  331 


SQUARE   ROOT. 

698.  The  Square  Root  of  a  number  is  one  of  the  two  equal 
factors  of  a  number.     Thus,  the  square  root  of  25  is  5.     5  x  5  =  25. 

699.  To  find  the  square  root  of  a  number. 

700.  RULE. — Beginning    at  units*  place,  separate  the 
given  number  into  periods  of  two  figures  each. 

Find  the  greatest  square  in  the  left-hand  period,  and 
write  its  root  at  the  right  in  the  form  of  a  quotient  in  divis- 
ion. Subtract  this  square  from  the  left-hand  period, 
and  to  the  remainder  annex  the  next  period  to  foi*m  a 
dividend. 

Double  the  part  of  the  root  already  found  for  a  trial 
divisor.  Find  how  many  times  this  divisor  is  contained  in 
the  dividend,  exclusive  of  the  right-hand  figure,  and  write 
the  quotient  as  the  next  figure  of  the  root.  Annex  this  quo- 
tient to  the  right  of  the  trial  divisor  to  form  the  complete 
divisor.  Multiply  the  complete  divisor  by  the  last  figure  of 
the  root,  and  subtract  the  product  from  the  dividend. 

To  the  remainder  annex  the  next  period,  and  proceed  as 
before. 

NOTE. — When  the  given  number  is  a  decimal,  separate  the  number  into 
periods  of  two  figures  each,  by  proceeding  in  both  directions  from  the  decimal 
point. 

EXAMPLES. 

701.  1.  Find  the  square  root  of  1156. 

OPERATION.  ANALYSIS. — Beginning  at  units'  place,  separate  the  num- 

11 '5  6  (  34     ber  into  periods  of  two  figures  each.     The  greatest  square 
9  in  the  left-hand  period  (11)  is  9,  and  the  root  is  3,  which  is 


64 


256 


written  in   the   quotient.      By  subtracting  this  square   (9) 


from  the  left-hand  period  (11)  and  annexing  to  the  remain- 
der (2)   the  next   period  (56),  we    form  the  dividend,  256. 


0  By  taking  twice  the  root  already  found  (3),  we  have  6  as  a 

trial  divisor,  which  is  contained  in  the  dividend  (25),  exclu- 
sive of  the  last  figure,  4  times.  "Write  4  in  the  quotient,  and  also  to  the 
right  of  the  trial  divisor,  forming  the  complete  divisor,  64.  Multiplying 
the  complete  divisor,  64,  by  4,  the  last  figure  of  the  root,  and  subtracting 
the  product  (256)  from  the  dividend  (256),  there  is  no  remainder.  34  is  the 
required  root. 


332  A  P  P  E  ND  I X.  [Art.  7O1. 

Find  the  square  root  of 


2.  1089. 
3.  14641. 
4.  18225. 
6.  46656. 

6.  47524. 
7.  65025. 
8.  86436. 
9.  97344. 

10.  119025. 
11.  1406.25. 
12.  512656. 
18.  232.5625. 

Z£.  3976036. 
15.  431.8084. 
16.  7463824. 
17.  387420489 

18.  Find  the  square  root  of  fff .     Of  f|f|.     Of 

NOTE. — In  finding  the  square  root  of  a  fraction,  extract  the  square  root  of 
the  numerator  and  denominator  separately. 

19.  Find  the  side  of  a  square  field  whose  area  is  412164  square 
rods. 

20.  Find  the  side  (in  feet)  of  a  square  whose  area  is  one  acre. 

21.  Extract  the  square  root  of  2  to  5  decimal  places. 


CUBE     ROOT. 

702.  The  Cube  Root  of   a   number   is   one   of   the   three 
equal  factors  of  that  number.     Thus,  the  cube  root  of  8  is  2. 
2x2x2  =  8. 

703.  To  find  the  cube  root  of  a  number. 

704.  RULE. — Beginning   at   units'  place,    separate  the 
given  number  into  periods  of  three  figures  each. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write 
its  root  at  the  right  in  the  form  of  a  quotient  in  division. 
Subtract  this  cube  from  the  left-hand  period,  and  to  the 
remainder  annex  the  next  period  to  form  a  dividend. 

Multiply  the  square  of  the  root  already  found  by 
300  for  a  trial  divisor.  Find  how  many  times  this 
divisor  is  contained  in  the  dividend,  and  write  the  quo- 
tient as  the  next  figure  of  the  root.  Add  to  the  trial 
divisor  thirty  times  the  product  of  the  last  figure  of  the 
root  and  the  other  figures  of  the  root,  and  the  square  of 
the  last  figure  to  form  the  complete  divisor.  Multiply 
the  complete  divisor  by  the  last  figure  of  the  root,  and 
subtract  the  product  from  the  dividend. 

To  the  remainder  annex  the  next  period,  and  proceed 
as  before. 


Art.  705.] 


CUSS    ROOT, 


333 


E  X.A  M  P  L  E  S  . 


7O5.  1.  Find  the  cube  root  of  39304. 


Trial  divisor.         300  X  32=r2700 

30x3x4=  360 
4«=     16 

Complete  divisor,     .    .    . 


3076 


39'304 

27 


34 


12  304 


12304 


OPERATION.  ANALYSIS.— Beginning 

at  units'  place,  separate 
the  number  into  periods 
of  three  figures  each. 
The  greatest  cube  in  the 
left-hand  period  (39)  is 
27,  and  its  root  is  3, 
which  is  written  in  the 
quotient.  By  subtract- 
ing this  cube  (27)  from 
the  left-hand  period  (39),  and  annexing  to  the  remainder  (12)  the  next  period 
(304),  we  form  the  dividend,  12304.  By  multiplying  the  square  of  the  root 
already  found  by  300  (multiply  by  3  and  add  two  ciphers),  we  form  the 
trial  divisor,  2700  (300  x  32  =  2700).  The  trial  divisor,  2700,  is  contained  in 
the  dividend,  12304,  4  times.  Write  4  as  the  next  figure  of  the  root.  To 
form  the  complete  divisor,  add  to  the  trial  divisor  (2700)  30  times  the  prod- 
uct of  the  last  figure  (4)  of  the  root  and  the  other  figure  (3)  (30  x  3  x  4=360), 
and  the  square  of  the  last  figure  (42  =  16).  Multiplying  the  complete  di- 
visor, 3076,  by  4,  the  last  figure  of  the  root,  and  subtracting  the  product 
(12304)  from  the  dividend  (12304),  there  is  no  remainder.  34  is  the  required 
cube  root. 

NOTE. — When  the  given  number  is  a  decimal,  separate  the  number  into 
periods  of  three  figures  each,  by  proceeding  in  both  directions  from  the 
decimal  point. 

Extract  the  cube  root  of 

2.  5832.  6.  551368.  10.  98611128. 

3.  10.648.  7.   7529536.  11.  279.726264. 

4.  314432.  8.  9.663597.  12.  435519.512. 
£.•474552.  9.  13651.919.  13.  676836152. 

14.  The  product  of  three  equal  numbers  is  551368.     What  are 
the  numbers  ? 

15.  A  cubical  block  of  marble  contains  103823  cubic  inches. 
What  is  its  length  ? 

16.  Find  the  cube  root  of  •§  £}f     Of  AVVV     Of  ifMf 

NOTE. — In  finding  the  cube  root  of  a  fraction,  extract  the  cube  root  of  the 
numerator  and  denominator  separately. 

17.  A  cubical  box  holds  20  bushels.     Find  the  length  of  its 
side. 


MENSURATION. 


706.  To  find  the  area  of  a  rectangle  (319)  or  paral- 
lelogram. 

707.  RULE. — Multiply  the  length  by  the  width.     (32O) 

708.  To  find  the  area  of  a  triangle. 

709.  RULE. — Multiply  the  base  by  half  the  perpendicu- 
lar height. 

710.  To  find  the  area  of  a   triangle  when  the  three 
sides  are  given. 

711.  RULE. — From  half  the  sum  of  the  three  sides  sub- 
tract each  side  separately.     Then  multiply  the  half  sum 
and  the  three  remainders  together,  and  extract  the  square 
root  of  the  continued  product. 

712.  To  find  the  area  of  a  trapezoid. 

713.  RULE. — Multiply  half  the  sum  of  its  parallel  sides 
by  the  perpendicular  distance  between  them. 

714.  A  Trapezoid  is  a  four-sided  figure  having  only  two  of 
its  sides  parallel. 

715.  To  find  the  area  of  a  circle. 

716.  RULE. — Multiply  the  square  of  the  diameter  by 
.7854 ;  or  multiply  the  square  of  the  radius  by  S.lJj-16. 

717.  To  find  the  surface  of  a  sphere. 

718.  RULE. — Multiply   the   square   of  the   diameter  by 
3.1416. 

719.  To  find  the  solid  contents  of  a  prism  or  cylinder. 

720.  RULE.—  Multiply  the  area  of  the  base  by  the  height 
or  length  of  the  prism  or  cylinder. 

721.  To  find  the  solid  contents  of  a  sphere. 

.  RULE. — Multiply    the  cube    of    the    diameter    by 


Art.  723.] 


-V E N  S  UR  ATIO  N. 


335 


723.     GEOMETRICAL,    CONSTANTS. 


CIRCLES. 


Circumference  = 
Circumference  = 
Radius  = 

Radius 

Radius  = 

Radius  = 

Diameter  = 

Diameter  = 
Diameter 

Diameter  = 

Area  = 

Area  = 
Side  of  inscribed  square 
Side  of  equal  square 


Diameter  x  3.1416. 
Radius  x  6.2832. 
Circumference  -f-  6.2832. 
Circumference  x  .1592. 
A/Area"  -^3.1416. 

A/Area  x  .5642. 
Circumference  -f-  3.1416. 
Circumference  x  .3183. 


=   A/Area -f-.  7854. 
=   VArea  x  1.1284. 
=  Radius  square  x  3.1416. 
=  Diameter  square  x  .7854. 
=  Diameter  x 


,7071. 


=  Diameter  x  .8862. 


Side  of  inscribed  equilateral  triangle  =  Diameter  x  .8603. 


Surface 

Surface 

Volume 

Volume 

Diameter 

Diameter 

Radius 

Radius 

Circumference 

Circumference 

Side  of  inscribed 

Side  of  inscribed 


SPHERES. 

=  Diameter  square  x  3.1416. 

=  Circumference  square  x  .3183. 

=  Diameter  cubed  x  .5236. 

=  Circumference  cubed  x  .0169, 

=  A/Surface  x  .5642. 

=  ^Volume  x  1.2407. 

=  A/Surface  x  .2821. 

=  ^Volume  x  .6204. 

=  A/Surface  x  1.7725. 

=  ^Volume  x  3.8978. 

cube  =  Diameter  x  .5774. 

cube  =:  Radius  x  1.1547. 


ROOTS,     ETC. 

Diagonal  of  square  =  Side  x  1.4142. 

Square  root  of  2        =  1.4142. 

Side  of  square  =  Diagonal  x  .7071. 


APPENDIX.  [Art.  721. 


EXAMPLES. 

1.  What   is   the   area  of   a   rectangular   piece   of   land 
15.50  chains  long  and  12.25  chains  wide  ? 

2.  Find  the  area  of  a  triangular  piece  of  land  whose  base  is 
3.25  chains  and  perpendicular  width  5.20  chains. 

3.  What  is  the  area  of  a  triangular  piece  of  ground  whose 
sides  are  20,  30,  and  40  rods  respectively  ? 

4'  A  piece  of  land  in  the  form  of  a  trapezoid  has  two  parallel 
sides,  1225  ft.  and  750  ft.  respectively,  and  the  perpendicular 
distance  between  them  is  1540  ft.  What  is  its  area  ? 

5.  The  circumference  of  a  circular  race-course  is  one  mile. 
What  is  its  diameter  in  feet  ? 

6.  Find  the  circumference  of  a  circle  whose  diameter  is  150 
feet. 

7.  A  cow  is  tied  to  a  stake  by  a  rope  42  feet  long.     Upon  how 
much  surface  can  she  graze  ? 

8.  The  diameter  of  a  globe  is  2G  inches.     What  is  the  area  of 
its  surface  ? 

9.  How   many   gallons   of  water   will   a   cistern  hold  whose 
diameter  is  10  ft.  and  depth  6  ft.  ? 

10.  What  are  the  solid  contents  of  a  globe  whose  diameter  is 
26  inches  ? 

11.  A  lake,  whose  diameter  is  1000  ft.,  is  covered  with  ice  8  in. 
thick.     What  is  the  weight  of  the  ice  in  tons,  if  a  cubic  ft.  of  ice 
weighs  920  oz.  avoirdupois  ? 

12.  Find  the  length  of  the  side  of  a   cubical  bin,  whose  con- 
tents are  100  bushels. 

13.  The  area  of  a  square  field  is  5  acres.     What  is  the  ?ength 
of  a  side  ? 

14.  Find  the  solid  contents  of  a  log  24:  ft.  long  and  2ft.  in 
diameter. 

15.  In  making  a  square  pond  whose  side  was  204  ft.,  10000 
cubic  yards  of  earth  were  taken  out.     What  was  its  depth  ? 

16.  A  room  is  13  ft.  4  in.  by  13ft.  6  in.     How  many  yards  of 
carpet,  f  of  a  yard  wide,  will  cover  it  ? 

17.  Find  the  diameter  and  circumference  (in  rods)  of  a  cir- 
cular field  containing  10  acres. 

18.  How  many  acres  in  a  field  one  mile  in  diameter  ? 


GENERAL    AVERAGE. 


725.  If,  in  time  of  danger  or  distress,  any  loss  or  expense  is 
voluntarily  incurred  for  common  safety  of  vessel,  freight,  and 
cargo,  such  loss  or  expense  is  made  good  by  a  "  General  Aver- 
age ; "  the  amount  or  value  of  such  loss  or  expense  being  assessed 
upon  the  value  of  all  interests  involved  and  benefited. 

All  other  losses  and  expenses  are  of  a  "  Particular  Average  " 
nature,  and  are  to  be  borne  by  the  specific  interests  to  which  they 
apply. 

726.  The  losses  and  expenses  constituting  general  average  are 
as  follows  : 

1.  Jettison,   or  throwing  overboard   of   cargo  to  lighten  the 
ship ;  damage  to  cargo  by  water  going  down  the  hatches  during 
jettison ;  damage  by  chafing  or  breaking  after  jettison ;  freight  on 
cargo  jettisoned. 

2.  Sacrificing  ship's  materials,  as  the  cutting  away  of  masts, 
spars,  etc.     One-third  of  the  cost  of  repairs  of  ship's  materials  is 
a  special  charge  on  the  ship,  as  the  new  work  is  considered  better 
than  the  old.     No  deduction  is  made  for  anchors. 

3.  Expense  of  floating  a  stranded  ship. 

4.  Expense   of    entering   a  port   of   refuge,  either  to  repair 
damage  which  renders  it  dangerous  to  remain  at  sea,  whether  such 
damage  were  caused  by  accident  or  sacrifice  ;  or  otherwise  to  avert 
a  common  danger. 

5.  Expense  of  discharging  cargo  for  the  purpose  of  making 
repairs,  warehouse  rent,  reloading  cargo,  outward  expenses,  etc. 

6.  Wages  and  provisions  of  crew  from  the  date  of  bearing  up 
until  ready  for  sea. 

737.  Contributory  Interests  and  Values.  —The  ship  con- 
tributes on  its  full  value  at  the  time  which  is  made  the  basis  of 
contribution. 


338  GENERAL      AVERAGE.  [Art.  727. 

The  cargo  contributes  on  its  net  market  value  at  the  port  of 
destination,  less  freight  and  charges  saved. 

The  freight  contributes  on  the  full  amount.,  less  -J-  for  the 
wages,,  etc.,  of  crew.  In  the  States  of  New  York,  Virginia,  Cali- 
fornia, and  some  others,  |-  is  deducted. 

The  underwriters  (Insurance  companies)  contribute  to  the  general  average 
such  a  part  of  the  expense  as  the  insured  value  is  of  the  market  value  of  the 
goods  (542).  If,  for  example,  a  cargo  is  insured  for  $10000  and  is  worth  in 
the  market  $12000,  the  underwriters  are  liable  to  pay  f  of  the  general  average 
expense. 

728.  To  give  rise  to  general  average,  it  must  be  shown  that 
there  was  an  imminent  common  danger,  that  the  sacrifice  was 
voluntary  and  necessary,  and  that  the  act  was  prudent  and  suc- 
cessful. 

729.  An  Average  Adjuster  is  a  person  who  is  familiar  with 
the  general  average  laws  of  the  leading  commercial  nations,  and 
who  adjusts  and  apportions  the  losses  and  expenses  of  a  general 
average. 

The  principal  difficulty  of  an  adjuster  is  to  decide  whether  the  loss  should 
be  made  good  by  a  general  average  or  should  be  made  a  special  charge  (par- 
ticular average)  upon  some  particular  interest.  After  the  general  average 
charges  are  determined,  the  apportionment  of  the  loss  among  the  several  con- 
tributory interests  is  a  simple  arithmetical  problem. 


EXAM  PLES. 

73O.  1.  The  bark  Liberty  sailed  from  New  York  for  Galves- 
ton  with  the  following  cargo  :  Shipped  by  A,  $5600  ;  by  B,  $8700  ; 
by  C,  $16308 ;  by  D,  $8360.  After  two  clays  out  the  bark  en- 
countered heavy  gales  and  was  damaged  to  the  amount  of  $630. 14. 
On  the  fifth  day  the  vessel  began  to  take  water,  and  for  the  safety 
of  the  vessel  and  the  cargo  the  bark  bore  away  for  New  York  for 
repairs.  The  disbursements  of  the  agent  at  New  York  were  as  fol- 
lows :  Custom-house  fees,  pilotage,  protest,  towage,  unloading  and 
reloading  cargo,  wharfage,  inspection,  consul  fees,  $1369.43  ;  bill 
of  H.  Robin  &  Co.,  shipwrights,  etc.,  $436  ;  bill  of  Joseph  Patti, 
ceiling  ship,  $194.14.  Agent's  commission  for  advancing  funds 
and  paying  above  bills,  5%  ;  on  value  of  cargo  landed,  $17388,  \\%. 
Wages  and  provisions  of  seamen  from  point  of  deviation,  $630.47. 
The  gross  freight  was  $8096,  and  seamen's  wages,  etc.,  \  of  gross 


Art.  730.] 


GENERAL     AVERAGE. 


339 


freight.  How  is  the  settlement  to  be  made,  the  value  of  the  ship 
being  $10000  and  the  adjuster's  fee  $100  ? 

NOTES. — 1.  In  a  general  average,  extracts  from  the  log  of  the  ship,  the 
testimony  of  its  officers,  a  complete  statement  of  all  expenses  incurred,  with 
the  vouchers  for  the  same,  and  all  papers  having  any  bearing  upon  the  case 
are  presented  to  the  adjuster.  The  total  amount  of  each  item  is  entered  in  a 
column  at  the  left  of  his  statement  of  charges,  and  the  amount  is  also  entered 
in  its  proper  column  at  the  right.  In  addition  to  the  general  average  column, 
there  are  usually  columns  to  the  right  for  the  special  charges  upon  the  ship, 
owners,  or  cargo. 

2.  After  determining  the  general  average  loss,  divide  it  among  the  con- 
tributory interests  in  proportion  to  their  values,  by  any  of  the  methods  given 
in  Ex.  13,  page  293. 

STATEMENT  OF  CHARGES. 


General 

Ship  and 

Average. 

Owners. 

1369 

43 

Expense  of  entering  harbor,  landing  cargo,etc. 

1369 

43 

436 

Bill  of  H.  Robin  &  Co.,  shipwrights,  etc. 

436 

194 

14 

"     "  Joseph  Patti,  ceiling  ship. 

194 

14 

Agent's  commission  for  advancing  funds  and 

99 

98 

paying  above  bills,  5%. 

68 

47 

31 

51 

Agent's  commission  on  value  of  cargo  landed, 

217 

35 

$17388,  \\%. 

217 

35 

630 

47 

Wages,  etc.,  of  seamen. 

630 

47 

100 

Adjuster's  fee. 

100 

General  average. 

2385 

72 

3047 

37 

Ship  and  owners. 

661 

65 

CONTRIBUTORY  INTERESTS  AND  APPORTIONMENTS  IN  GENERAL 

AVERAGE. 


Ship,   value  10000  @.  .045  pays 

450 

Freight,  8098 

Less£,  4048   4048  ©  .045 

182 

16 

Cargo, 

A,    5600        @  .045   " 

252 

B,    8700        ®  .045   " 

391 

50 

C,    16308        @  .045   " 

733 

86 

D,    8360        @  .045 

376 

20 

38968  @  .045   " 

1753 

56 

53016  @  .045   " 

2385 

72 

$2385.72  -r-  $53016  =  .045. 


340 


APPENDIX. 
SETTLEMENT 


[Art.  13O. 


BALANCES. 

DR. 

CB. 

To  pay. 

To  receive. 

Vessel  and  Owners. 

Pay  ship's  proportion  of  Gen.Aver. 

450 

"    freight's      " 

182 

16 

"    owner's  column. 

661 

65 

Receive  seamen's  wages. 

630 

47 

663 

34 

Cargo. 

Pay  proportion  of  Gen.  Average. 

1753 

56 

1753 

56 

Agents  of  Vessel. 

Receive  their  disbursements. 

1999 

57 

"          "     commission. 

317 

33 

2316 

90 

Adjusters. 

Receive  their  fee. 

100 

100 

3047 

37 

3047 

37 

2416 

90 

2416 

90 

2.  The  general  average  charges  were  $4375.86,,  and  the  con- 
tributory interests  $64325.    What  was  the  per  cent,  of  loss  ?    What 
was  the  loss  of  Mr.  B.,  whose  goods  were  valued  at  $7250  ? 

3.  Suppose  A's  goods  in  Ex.  1  were  insured  for  $5000,  how 
much  of  the  loss  would  be  shared  by  the  insurance  company  ? 

4.  The  ship  Amazon,  from  Aspinwall  to  New  York,  being  in 
distress,  threw  overboard  part  of  the  cargo,  cut  away  the  masts, 
and  finally  bore  away  to  a  port  of  refuge  to  repair  in  order  to  com- 
plete the  voyage.     The  cost  of  replacing  masts  and  rigging  cut 
away  was  $6000   (less  |  new  for  old) ;  the  cargo  jettisoned  was 
worth  compared  with  sound  cargo  delivered  at  destination  $2000 ; 
freight  on  cargo  jettisoned,  $200  ;  expenses  of  entering  port  of 
refuge,  discharging,  storing  and  reloading  cargo,  $1000;  wages  of 
master  and  crew  from  time  of  bearing  away  until  ready  for  sea, 
$600;  provisions  of  master  and  crew  for  same  time,  $500;  adjuster's 
fee,  $100.     The  vessel  was  valued  at  destination  at  $20000  (deduct 
gross  repairs  and  add  amount  made  good) ;  cargo,  value  on  arrival, 
$40000  (add  amount  made  good) ;  freight  collected,  $4000  (add 
amount  made  good  and  deduct  J).     What  was  the  per  cent,  of 
loss,  and  how  was  the  settlement  made  ? 

5.  The  cargo  of  the  ship  Amazon  was  insured  for  $36000.    How 
much  was  the  claim  against  the  insurance  company  ? 


Art.  73O.]  GENERAL     AVERAGE.  341 

6.  The  ship  Union,  in  her  passage  from  Liverpool  to  Boston, 
during  a  storm  threw  overboard  cargo  to  the  amount  of  $1580, 
and  cut  away  masts  and  rigging.     She  then  entered  the  port  of 
Halifax  for  repairs.     The  cost  of  replacing  the  masts  and  rigging 
which  were  voluntarily  sacrificed,  was  $4578  (less  £  new  for  old)  ; 
cost  of  repairing  accidental  damage,  $568  ;  freight  on  cargo  jetti- 
soned, $314. 75  ;  expense  of  entering  port  of  refuge,  discharging 
cargo,  etc.,  $716.87;  wages  and  provisions  of  crew,  $608 ;  adjuster's 
fee,  $150.     The  value  of  vessel  on  arrival  at  Boston  was  $30000 
(deduct  gross  repairs  and  add  amount  made  good) ;  value  of  cargo 
delivered,  less  freight  and  duty,  $48475  (add  amount  jettisoned)  ; 
total  expected  earning  of  freight,  $16320  (less  £  in  Boston.    See 
Art.    727).     The  cargo  was  shipped  by  the  following  persons : 
A  $8519,  B  $20376,  C  $6875,  and  D  $14285.    The  cargo  jettisoned 
was  a  part  of  A's  shipment.     How  ought  the  settlement  to  be 
made? 

7.  The  ship  Ocean  Qmen,  from  Pernambuco  to  New  York, 
sprang  a  leak  off  Cape  St.  Koque,  and  for  the  safety  of  the  vessel 
and  cargo,  threw  overboard  part  of  the  cargo  and  put  into  Maran- 
ham  for  repairs.     The  disbursements  at  Maranham  by  the  master 
of  the  vessel,  including  commissions,  were  as  follows  :  Expenses  of 
entering  harbor,  discharging,  storing,  and  reloading  cargo,  $648.75 ; 
caulking  and  painting  ship,  carpenter  work,  etc.,  $843.    Value  of 
cargo  delivered  at  New  York,  $34310.24;    of  cargo  jettisoned, 
$1580.76 ;  freight  on  cargo  jettisoned,  $364  ;  wages  and  provisions 
of  crew,  $304 ;  adjuster's  fee,  $150 ;  agent's  commission  for  col- 
lecting amount  in  general  average,  2J%.     How  shall  the  settle- 
ment be  made,  if  the  net  value  of  the  ship  was  $3157  (value  on 
arrival  $4000,  less  repairs  $843),  and  the  total  expected  earning 
of  freight  was  $2516  (less  £)  ? 

8.  A  vessel  which  put  into  a  port  of  refuge  for  repairs  was 
without  funds.     It  being  very  difficult  to  obtain  a  loan  on  bot- 
tomry, or  to  negotiate  a  draft  on  the  owners  of  the  vessel,  the  mas- 
ter was  obliged  to  sell  part  of  the  cargo  to  raise  funds.    Value  of 
cargo  sold  compared  with  cargo  delivered  at  destination,  $4566.06 ; 
produced  at  sale,  $2985.30  ;  freight  on  cargo  sold  compared  with 
freight  on  cargo  delivered,  $363.93.    What  was  the  cost  of  funds, 
and  how  much  should  be  apportioned  to  each  interest,  the  general 
average  charges  being  $773.52,  the  special  charges  on  ship  $956.10, 
and  on  the  owners  $1181.06  ? 


343 


APPENDIX. 


[Art.  731. 


731.   FOREIGN  WEIGHTS  AND  MEASURES. 


ARGENTINE  CONFEDERATION. 
Metric  system  used   in  the  assess- 
ment of  duties.     Old  Spanish  weights 
and  measures  (See  Spain)  in  common 
use. 

AUSTRIA,  (AS  GERMANY.) 
BELGIUM,  (METRIC  SYSTEM.) 

BOLIVIA. 

The  metric  system  is  the  legal  sys- 
tem, but  the  law  has  not  been  rigidly 
enforced.  Old  Spanish  weights  and 
measures  (see  Spain)  still  in  use.  For 
coin  weight  the  metric  gram  is  used. 

BRAZIL,  (METRIC  SYSTEM.) 
Diamonds  are  permitted  to  be  sold 

according  to  the  old  Portuguese  outava 

(55.34  grains). 
Ships'   freights    are    for  the    most 

part,  settled  according  to  the  English 

ton  (2240  lb.). 

CANADA,  (AS  GREAT  BRITAIN.) 


CHILI,  (AS  BOLIVIA.) 
For  custom    purposes    the   metric 
system  is  enforced. 


1  Tael 
1  Catty 
1  Picul 
1  Chih 
1  Chang 


CHINA. 

=     1£  oz.  av. 
=     H  U.  av. 
=     133£  Ib.  av. 
=     14.1  inches. 
=     11.75  feet. 


COLUMBIA,  (METRIC  SYSTEM.) 

DENMARK. 

1  Pound  (I  kilogram)  =  1.102  Ib.  av. 
1  Centner  (100  Ib.)     =  110.23  Ib.  av. 
1  To'nde  of  grain       =  3.948  U.  S.  bu. 
1  To'nde  of  coal         =  4.825  U.  S.  bu. 


1  Fod  (Foot)  =  1.03  U.  S.  ft. 

1  Viertel  =  2.04  U.  S.  gal. 

1  Alen  (Ell)  =  .6864  yd. 

Coinage  laws  are  metric.  The  in^ 
troduction  of  complete  metric  system 
is  in  prospect. 

ECUADOR,  (METRIC  SYSTEM  ) 
EGYPT,  (METRIC  SYSTEM.) 

FRANCE,  (METRIC  SYSTEM.) 
The  old  French   aune  =  1^  yd.   is 
still  used  to  some  extent  in  the  silk 
industries  of  France  and  the  U.  S. 

GERMANY. 

Metric  system  with  a  few  changes 
in  subdivisions  in  general  use. 
1  Pound  (£  kilogram)  =  1.1023  Ib.  av. 
1  Centner  (100  pounds)  =  110.23  Ib.  av. 
1  Wispel  (metric  ton)  =  2204. 6  Ib.  av. 


GREAT  BRITAIN. 

1  Imp.  Gallon  =  1.2  U.  S.  gal. 

1     "     Bushel  =  1.03  U.  S.  bu. 

1     "     Quarter  =  8.25  U.  S.  bu. 

1  Ale  or  Beer  Gallon  =  1 .22  U.  S.  gal. 

1  Cental  =  100  Ib. 

1  Quarter  of  Wheat  )       XQn  „ 
at  London  f  =     *°  lb' 

1  Quarter  of  Wheat  at  Hull  )        KA/(  „ 
and  Newcastle.  \=     **• 

1  Quarter  of  Wheat  at  Dun- 
dee and  other  places. 
Metric  system  permitted  by  law  of 

1864. 

GREECE. 

Metric  system  with  the  common 
Grecian  names  in  general  use. 

In  the  Ionian  Islands  the  English 
weights  and  measures  have  been 
legalized  since  1829. 


496  lb. 


Art.  731.]  FOREIGN    WE  IGIITS  AND    MEASURES. 


343 


INDIA. 

1  Seer  =  16  chattucks. 
1  Bombay  Maund  of  40  seers: 


1 

1  Surat 
1      " 
1      " 


42 
40 
42 
44 


=29.4' 


1  Bengal  Factory  Maund  =74f  " 
1  "  Bazaar  "  =82£  " 
1  Madras  Maund  =25  " 

1  Bom'yCandyof20Maunds=560  " 
1  Surat  "  "  "  =746f" 

1  Madras  "  "  "  =500  " 
1  Travancore  "  "  "  =660  " 
1  Tola  =180  gr. 

1  Guz  of  Bengal  =1  yard. 

1  Gorge  =20  units. 

1  Gorge  Pound  =20  Ib. 

Metric  system  permissive. 

ITALY. 

1  Palm  =  .555  cu.  ft. 
Metric  system  in  general  use. 

JAPAN. 

1  Picul  =  133£  Ib.  av. 
For  coinage,  in  part,  the  metric  unit 
of  weight  is  used. 

JAVA. 

1  Amsterdam  Pond  =  1.09  Ib.  av. 
1  Picul  =  133i      " 

1  Catty  =  li 

1  Chang  =  4  yards. 

MEXICO. 

Weights  and  measures  are  legally 
the  metric,  but  the  metric  system  is 
not  generally  in  force,  the  old  Spanish 
weights  and  measures  (see  Spain)  being 
still  employed. 

NETHERLANDS. 

Metric    system    with  a  change  in 
names  in  general  use. 
1  Last  (30  hectoliters)  =  85.134  bu. 

NORWAY  AND  SWEDEN. 
1  Swedish  Skalpond      =  0.93£  Ib.  av. 
1  Swedish  Centner        =934-        " 


1  Norwegian  Pund        =  1.1  Ib.  av. 
1  Swedish  Fot  =  11.7  inches. 

1  Norwegian  Fod  =  12.02      " 

In  Norway  the  metric  system  is  used 
to  some  extent. 

In  Sweden,  the  coin  weight  and  the 
medicinal  and  apothecary  weight  are 
metric.  The  complete  metric  system 
has  been  obligatory  since  1882. 

PORTUGAL. 

Metric  system  compulsory  since  Oct. 
1,  1868. 

The  chief  old  measures  are — 
1  Libra  =1.012  Ib.  av. 

1  Almunde  of  Lisbon  =4.42  U.  S.  gal. 
1  Alquiere  =.3928  U.  S.  bu. 

EUSSIA. 

1  Pound  =0.9  Ib.  av. 

1  Pood  (63  to  a  ton)     =36 
1  Berkowitz  =360      " 

1  Chetvert  =5.956  U.  S.  bu. 

1  Vedro  =3.25  U.  S.  gal 

1  Arsheen  =28  inches. 

1  Ship  Last  =2  tons. 

Metric  system  partially  in  use. 

SPAIN,  (METRIC  SYSTEM.) 
In  many  of  the  South  American 
States  and  in  Cuba,  the  old  Spanish 
weights  and  measures,  principally 
Castilian,  are  used.  They  are  as  fol- 
lows : 

1  Libra  =  1.014  Z&.  av. 

1  Arroba  (25  Libras)  =  25.36  " 
1  Quintal  (100  Libras)  =101.44  " 
1  Vara  =  .914  yd. 

SWITZERLAND. 

Metric  system  used  with  some 
changes  of  names  and  subdivisions. 
Pure  metric  system  optional. 

TURKEY,  (METRIC  SYSTEM.) 

URUGUAY,  (AS  ARGENTINE  CONFED- 
ERATION.) 

VENE3UBLA,   (METRIC  SYSTEM.) 


DETECTION    OF    ERRORS 


TRIAL    BALANCES. 

732.  The  following  hints  apply  to  the  detection  of  errors  in 
trial  balances,  or  in  any  operation  in  which  errors  are  made  in 
addition  or  subtraction,  or  in  transferring  numbers  from  one  place 
to  another. 

1.  Ascertain  the  exact  amount  of  the  error.     Much  time  is 
sometimes  wasted  in  looking  for  errors  which  do  not  actually  exist. 

2.  Kevise  carefully  the  additions  of  the  trial  balance  before 
looking  for  the  error  in  the  ledger  or  other  books. 

3.  If  the  error  is  in  one  figure  only  (as  2000,  100,  50,  etc.),  it 
is  probably  an  error  in  addition  or  subtraction. 

4.  If  an  amount  is  entered  on  the  wrong  side  of  an  account,  or 
is  added  when  it  should  be  subtracted  or  vice  versa,  the  error  will 
be  twice  the  amount. 

5.  If  the  digits  of  any  number  are  written  to  the  right  or  left 
one,  two,  or  three  places,  and  the  error  be  divided  by  9,  99,  or 
999  respectively,  the  quotient  will  be  the  number. 

Thus,  if  $427  be  written  $4.27,  the  error  will  be  $422.73  ;  which  divided 
by  90  (by  9  and  11),  the  quotient  will  be  $4.27. 

The  number  of  9's  by  which  the  number  can  be  exactly  divided  is  equal 
to  the  number  of  places  which  the  number  has  been  transferred  to  the  right 
or  the  left. 

6.  If  two  consecutive  digits  of  any  number  are  transposed,  the 
error  will  be  a  multiple  of  nine ;  and  the  quotient  obtained  by 
dividing  the  error  by  9  will  express  the  difference  between  the 
digits  transposed. 

Thus,  if  437,  be  written  473,  the  error  will  be  36 ;  which  divided  by  9 
produces  4,  the  difference  between  3  and  7.  The  same  error,  36,  will  arise  if 
the  figures  transposed  are  0  and  4, 1  and  5,  2  and  6,  4  and  8,  or  5  and  9. 

7.  If  the  error  contains  a  number  of  figures,  it  is  probable 
that  some  account  or  item  has  been  omitted. 

8.  Look  for  the  error  systematically,  and  not  in  certain  por- 
tions of  the  work  selected  at  random. 


ANSWERS. 


Art.  2O. 

3.  40865. 

5.  49374;  98748. 

77.  5,761,888; 

7.  1614. 

4.  110547. 

4.  H0775; 

195,249,432. 

2.  1654. 

5.  8495098. 

295400. 

70.  18,413,409; 

3.  19380. 

6.  853759. 

5.  243580; 

556,524.675. 

4.  23243. 

7.  999895. 

438444. 

13.  8,326,575; 

5.  26162. 

5.  1109975. 

6.  817281; 

173,434,110. 

6.  35130. 

9.  6419754. 

726472. 

14.  25,930,788; 

7.  4566. 

10.  72540. 

7.  130240; 

317,327,062. 

5.  3722. 

77.  57249251. 

182336. 

75.  93,309,006; 

9.  53609. 
10.  44601. 

70.  10648519. 
75.  113558829. 

5.  1,578,246; 
2,367,369. 

889,602,580. 
16.  4,428,648; 

77.  50480. 

14.  15562130. 

9.  494268; 

26,888,220. 

70.  34914. 

75.  74,299,273. 

617835. 

77.  744. 

16.  5,654,786. 

10.  4,690,158; 

18.  43200. 

Art.  2?. 

77.  90,119,023. 

7,035,237. 

79.  2,419,200. 

4.  4915.  9.  435. 

75.  122,882. 

77.  3,336,072; 

SO.  5250. 

5.  4857.  70.  508. 

79.  921294. 

2,919,063. 

07.  506880; 

6.  394.  77.  3642 

SO.  19.212,939. 

70.  4,072,384; 

1,098,240. 

7.  376.  70.  3645 

21.  4745. 

3,563,336. 

22.  26376. 

5.  321.  13.  3755 

00.  64535. 

75.  3,824,910; 

23.  106515; 

7^.  54877. 

S3.  45009. 

5,737,365. 

153720. 

75.  44444. 

24.  27369. 

14.  5,240,172; 

24.  576. 

16.  41568. 

05.  41976. 

3,742,980. 

25.  4608. 

77.  36311. 

26.  12464. 

75.  58080. 

S6.  5016. 

75.  84839. 

27.  62645. 

16.  2016. 

79.  139059. 

S8.  10514. 

77.  24256. 

Art.  47. 

20.  10078521. 

29.  3211. 

75.  63360. 

1.  144000; 

21.  561.  26.  3657. 

SO.  5821. 

79.  $194.40. 

1080000. 

00.  3691.  07.  6822. 

57.  4004. 

20.  1296. 

0.  138240; 

23.  1404.  05.  1711. 

50.  5038. 

864000. 

24.  7921.  09.  1440. 

33.  1235. 

Art.  44. 

3.  241920; 

25.  297.  30.  7529. 

54.  11,594,495. 

1.  63936;  831168. 

1,451,520. 

57.  14152. 

55.  193,941,760. 

0.  75218; 

4.  185500; 

50.  442,254,988. 

36.  $93,309,621. 

1,463,858. 

1,335,600. 

33.  433. 

57.  3025. 

3.  70272; 

5.  120000; 

54.  1771. 

55.  6558830. 

2,436,096. 

3,200,000. 

35.  4653. 

39.  3850814. 

4.  209387; 

6.  252000; 

55.  39247. 

40.  388904. 

1,915,125. 

1,036,000. 

57.  16098. 

41.  1106. 

5.  358661; 

7.  81600; 

55.  813210. 

A    m     f>  ** 

1,264,432. 

272000. 

59.  6399. 

Art.  35. 

6.  544375; 

5.  84000; 

40.  1,177,761,723. 

1.  1107.90. 

4,606,875. 

2,100,000. 

41.  5,302,516. 

0.  317.26. 

7.  720408; 

9.  9,680,000; 

4S.  324,423,840. 

5.  6622.70. 

7,213,316. 

67,760,000. 

43.  $9858.94. 

8.  661982; 

10.  18,500,000; 

44-  $419360.87. 

Art.  41, 

6.961,968. 

92,500,000. 

Art.  33. 

1.  164192; 
187648. 

9.  6,896,064; 
87,772,352. 

77.  7,407,000; 
44,442,000. 

1.  4337. 

0.  340236; 

10.  5,847,408; 

70.  11,760,000; 

0.  907823. 

226824. 

195,035,421. 

131,600,000. 

346 


AJVS  WERS. 


[Art.  47. 


13.  6,698,000; 

10.  88375;  94500. 

Art.  74. 

//.  921300; 

114,260,000. 

11.  95472;  96408. 

2.  7623;  7161. 

919450. 

14.  8,019,200; 

12.  91208;  92962. 

3.  8232;  7980. 

12.  868000; 

98,808,000. 

13.  77280;  80224. 

4.  6768;  6912! 

868875. 

15.  67,200,000; 

14.  15876;  14847. 

5.  35991;  386613. 

7&  838530; 

614,880,000. 

15.  40040;  41195. 

6.  39520;  413504. 

836836. 

16.  86,400,000; 

16.  82215;  80649. 

7.  49104;  523776. 

Art.  90. 

460,800,000. 
Art.  5O. 

17.  58422;  57876. 
Art.  62. 

Art.  78. 

1.  11872;  12432. 
2.  10506;  10608. 

3.  264;  176;  198; 
352;  473;  363; 
792;  891;  407; 
484;  1012; 
957;  1023; 
704;  385;  396; 
517;  187;  209; 
528;  627. 

2.  3724;  2964. 
3.  2523;  8613. 
4.  2655;  3105. 
5.  5928;  27768. 
6.  16653;  33733. 
7.  23925;  56925. 

Art.  65. 

1.  3136;  2304; 
4136. 
2.  4875;  3219: 
1656. 
3.  4851;  6375; 
816. 
4.  1739;  3819; 
3825. 
5.  8125;  4536; 

3.  13176;  12810. 
£  12412;  12992. 
5.  15515;  16240. 
6.  19536;  19008. 
7.  1,010,024; 
1,011,028. 
8.  1,134,000; 
1,138,500. 

2.  1206;  1809. 

12936. 

Art.  93. 

Art.  53. 

3.  1224;  1530. 

G.  5776;'  1296; 

'1.  10379;  1J0165. 

2,  2695;  3806; 

4.  4158;  4851. 

12996. 

£.  10752;  10304. 

3575;  4576; 

5.  6048;  6804. 

3.  10904;  11368. 

8624;  5687; 

6.  12420;  15525. 

Art.  81. 

4.  9828;  10692. 

9625;  10098; 

7.  10206:  40824. 

1.  7134;  7047; 

5.  91455;  93465. 

46398;  80564; 

8.  13986;  32634. 

4095. 

6.  95665;  97679. 

79398;  19008; 

5.  39096;  58644. 

2.  2068;  2912; 

7.  100188;  93104, 

48125;  92136. 

3306. 

S.  95692;  97728. 

Art.  68. 

3.  5548;  5925; 

Art.  56. 

2.  600;  900;  925: 

4284. 

^4r«.  99. 

2.  52704;  35424. 

1225;  1550; 

4.  1892;  2860; 

1.  39,456,174; 

3.  22227;  50907. 

9675;  11200; 

4221. 

26,304,116. 

4.  9387;  36207. 

12800;  18650; 

5.  13572;  11235: 

2.  24,413.116; 

5.  283745; 

10600;  20425; 

15250. 

I6,275,410f. 

260295. 

23425;  13600; 

3.  3,265,524; 

6.  378216; 

17925;  7950: 

Art.  84. 

2,721,270. 

600696. 

8100;  6400; 

2.  624;  7225; 

4.  19,517,701; 

7.  341284; 

13900;  230600; 

15616. 

11,152,972. 

174804. 

209450; 

3.  221;  9024; 

5.  2,057,613; 

8.  112875; 

132000;  43200; 

13216. 

1,371,742. 

150375. 

141200. 

4.  1224;  1221; 

6.  197,730,864; 

9.  85425;  42925. 

11024. 

123,581,790. 

10.  281869; 

Art.  7O. 

6.  625;  2021; 

7.  58,642,209; 

234969. 

3.  83763;  96929. 

21021. 

26.063,204. 

11.  338583; 

4.  126936; 

6.  1225;  3024; 

8.  178,606,127; 

386883. 

293088. 

24016. 

51,030,322. 

12.  75576;  338776. 

5.  43344;  42656. 

9.  49,377,285; 

13.  263375; 

6.  310148; 

-4?'£.  87. 

27,431,825. 

350875. 

137592. 

2.  9603;  9118. 

10.  31,025,988; 

7.  47775;  88725. 

5.  8008;  8360. 

24,131,324. 

Art.  59. 

8.  170556; 

4.  8277;  8544. 

11.  51,525,354$; 

2.  10205;  13345. 

863964. 

5.  7275;  7350. 

20,610,141$. 

8.  5292;  6048. 

9.  203912; 

6.  9016;  8556. 

12.  71,387,270; 

4.  7830;  9918. 

597376. 

7.  8084;  8170. 

35,693,635. 

5.  6768;  6016. 

10.  288834; 

<?.  985056; 

13.  106,315,682; 

6.  17276;  19744. 

399924. 

987042. 

60,75  l,818f. 

7.  37520;  42210. 

11.  138446; 

9.  981090; 

14.  41,152,263; 

8.  65664;  44928. 

394284. 

978120. 

15,432,  098|. 

9.  134147; 

12.  107880; 

10-  976108; 

15.  41,133,539; 

118365. 

186180. 

977090. 

22,436,475^. 

Art.  99.] 


ANSWERS. 


347 


16.  112,731,950;   I  21.  365^. 

5.  $25.12; 

14.  38. 

78,912,365. 

22.  138}|f. 

$1.227. 

15.  55. 

17.  57,447,290; 

S3.  677if. 

9.  53;  123. 

16.  23,  325  52. 

23,936,370^. 

24-  356|f. 

10.  414;  72. 

17.  32,  5,  43. 

18.  29,351,981f; 

25.  498299ff. 

11.  217;  36. 

18.  32,  52,  11. 

16,010,171^. 
19.  144,300,144f; 
1  12,233,445  1. 

26.  368T6.,V 
27.  153|fi. 

28.  786395^. 

12.  2304;  256. 
15.  1536;  256. 

14.  4088;  280. 

19.  22,  32,  5,  37, 
20.  24,  32,  17. 
21.  5s,  11,  31. 

30.  25,025,025; 

29.  21  61  ff  J. 

22  24  33  23 

16,683,350. 

30.  1519ilfff. 

-4ff.  139. 

25!  23'  32^  7,  19. 

SI.  4628  sq.  yds. 

51.  3,969,568. 

1.  4,964,629. 

24.  52,  7,  29. 

22.  10388s. 

32.  14960. 

2.  3,204,084. 

23.  89  doz.  ; 

$20.47. 

Art.  108. 

5.  1,042,916,880. 
4.  $4799.50. 

1.  30.   12.  360. 

24.  5280  ft. 

47;  66;  68; 

5.  5989  ft. 

2.  48.   15.  288. 

35.  20006  gal. 

115;  153;  198; 

6.  $18487. 

5.  120.  14.  2640. 

26.  10908  bu. 

152;  69;  71; 

7.  $11,014,811. 

4.  16.   15.  1260. 

73;  80. 

8.  92250. 

5.  84.   16.  240. 

Art.  1O3. 

9.  456. 

6.  360.  17.  660. 

1.  11840&; 

Art.  132. 

11.  $458. 

7.  504.  !<?.  3024. 

2152  iff. 

1.  $2837.46. 

12.  3582. 

<?.  60.   19.  360. 

S.  13100}f; 

S.  $1022.25. 

15.  132  acres. 

9.  120.  20.  840. 

999Uf. 

3.  $2775.87. 

14.  $2006. 

10.  480.  21.  9900. 

S.  9879f|; 

4.  $3383.08. 

15.  4664.24. 

11.  480.  22.  2520. 

looijyt. 

5.  $14.t)l. 

16.  $4.50. 

4.  10C28|f  ; 

6.  $18.51. 

17.  $46. 

Art.  169. 

4149* 

7.  $569.25. 

18.  $126.28. 

1.  1.     5.  14|. 

5.  16948&5 

8.  $1311.26. 

19.  $17377.65. 

2.  29-*f  .   6.  4|f. 

2037A. 

20.  $30,022,347.95. 

5.  41||f  7.  6T 

6.  170223}|; 

Art.  135. 

21.  $34,077,380. 

7142^4, 

1.  $97.44. 

22.  A,  $12283; 

Q   O 

7.  2014711; 

2.  $51. 

B,  $12568; 

10!  1008. 

2274^7g6y. 

5.  $273.53. 

C,  $12371; 

11.  $56. 

8.  97134f|; 

4.  $677. 

D,  $12071; 

12.  $0.51. 

16413ff4. 

5.  $3306. 

D,  lowest. 

l.f.  $9.37i. 

9.  72660||; 

6.  $187.60; 

S3.  339  head. 

14.  150  bu. 

10228fff. 

$276.48. 

24.  $9292.80. 

15,  750yd. 

10.  25052±f  ; 

7.  $86.73; 

25.  3000  Ib. 

16.  133  Ib. 

3161  fff.   • 

$939.12. 

26.  $1044. 

17.  $31.50. 

11.  11934811; 

8.  $400.80; 

27.  A,  $124; 

18.  36  cows. 

10663  |f  I. 

$2164.32. 

B,  $125. 

19.  243  bu. 

12.  17008||; 

9.  $3168; 

28.  $7.75. 

20.  1290  bu. 

8914i|f. 

$25344. 

29.  385231  +; 

21.  $16.92. 

13.  41933|f; 

10.  $1305; 

32102  +  . 

22.  $79. 

20093J&. 

$4241.25. 

25.  $21. 

14.  526009  &; 

11.  $2313.12; 

Art.  158. 

24.  56  Ib. 

57532-8g\. 

$13878.72. 

1.  32,  5,  7,  11. 

25.  66.   27.  90. 

15.  30366ig;' 

12.  $3756; 

2.  3,  7,  11,  13. 

26.  10.   •#£.  23/T, 

6642  Jf§. 

16.  58694/j-; 

$9615.36. 

3.  2,  33,  7,  11. 
4.  2,  32,  52,  7. 

Artil91. 

16261iff- 

Art.  138. 

5.  3,  52,  T2. 

1.  f  .    10.  j£. 

17.  116213||; 

1.  $3.45. 

6.  2,  3,  5,  7,  11. 

2.  |^.   11.  i  . 

114823k 

2.  15  Ib. 

7.  23,  32,  7,  13. 

5.  f  .    12.  f  . 

18.  39625||; 

5218m. 

5.  $.53;  $.24. 
4.  $19.04; 

8.  2,  7,  13,  43. 
9.  2,  3,  7,  11,  13. 

4.  f    13.  «. 

5.  y9^.   14-  IT- 

19.  50117ff  ; 

$13.60. 

10.  23,  11,  61. 

?;  i*-  ^'  If 

2584i||. 

5.  $48;  $36. 

11.  32,  52,  17. 

6.  $.87;  $.84. 

12.  22,  II3. 

8.  \.    17.  yVgV 

'  17274}||. 

7.  $1.48:  $.185. 

13.  213. 

5.  f  .  1*.  -B. 

348 


[Art.  191 


If ;  It;  If- 
e.  T7A;  *;  tfl 
7.  W;  ft;  ft- 
A  ft;  ft;  ft- 


*.  f*  H;W; 
If. 

4-11  W;tt; 


5.  4 


-  r,  .  66 

sr>  **• 


***- 


5. 

e.  ^F;  W;  *F. 
7.  H*;  W;  W- 
A  V;  W;  W- 


2.  $16. 
5.  93f ;  52. 

4.  69;  1251 

5.  57f ;  884. 

7  36*f  •  111 
8\  32l|;  78. 
9.  49|;  16J| 

10.  24A;  13f 

11.  20$i;  15| 

12.  27|; 


14.  64| 

is-  BH;  7f&. 

16.  22^;  62$. 

17.  18H;  22H. 


15.  463TV. 


m  704-. 
^1.  270f. 


^.  388H- 
25.  89^. 
^.  472TV. 
27.  88*. 
0£.  1263. 
^9.  2991. 
30.  1212. 


217 


22£  bu. 
3.  $364. 

£  $3684. 
5.  $21.22^. 


<<?.  2f. 

9.  1326. 

10.  1884. 

11.  745|. 

12.  440. 
15.  1379|. 

14.  3364|. 

15.  46154. 

16.  18024. 

17.  19381|, 

18.  5955. 

19.  36706J. 

20.  50407|. 
01.  2391f 
22.  13136|. 
25.  10158f. 

24.  14865. 

25.  77893f. 

26.  204837|. 

27.  2520371. 
05.  74571. 
29.  289790f. 

50.  322887. 

51.  389291$. 

32.  79032$. 

33.  259546|. 

34.  183288$. 

55.  486354^. 

56.  340184. 

57.  342468f. 
38.  413568f. 

Art.  221, 

1.  $2.18*. 

2.  $11.14f. 

3.  $36.651. 

4.  63. 

5.  72. 

6.  30f . 


7.  1216. 
5.  1287. 
9.  2010k 

10.  3105. 

11.  9328k 

12.  8500. 
15.  41974. 

14.  3091$. 

15.  4606|. 

16.  50570. 

17.  19017f. 

18.  53462$. 

19.  57734f. 

20.  70977f. 

21.  47078$. 

Art.  224. 

1.  A-   18.  3|. 
A-   ^.  ff. 


25.  1152 

26.  281  i 
756. 

<§.  625. 


227. 

1.  7441| ;  32612|. 
0.  7741^;  9031, 
5.  3845f ;  141  ( 

4.  73021^; 
16233|f. 

5.  82651 


6.  19531  A; 
41929|. 

7.  203614; 

28807H- 
A  17355^; 

18275TV. 
9.  16464f ; 

15234f. 
10.  26957^; 


Art.  231. 


0.  T5F. 

3.  ¥\. 

4.  A. 


6. 

7.  57$. 

5.  145ft 


Art.  231.] 


WERS. 


349 


9.  116ft. 

9.  303520£. 

59.  2520. 

75.  31.04581. 

70.  81TV 

70.  10. 

60.  $25,574,327.33. 

14.  53.391. 

77.  70f. 
12.  72fl. 
75.  214TV 

77.  12. 
72.  21. 
75.  1763f  ;  352|. 

Art.  256. 

1.  .50.   5.  .75. 

75.  68.0444. 
76.  59.702f. 
77.  5512.33f. 

14.  5761. 
75.  483f. 

14.  186.004^. 
75.  145fll. 

2.  .875.   4.  .375. 
5.  .4375. 

75.  931.057|. 
19.  .46571. 

76.  703f  . 

77.  $280.59. 

6.  .78125. 

77.  809T7¥. 

75.  $48492. 

7.  .425. 

Art.  268. 

75.  1593ft. 

19.  951  bu- 

5.  .661. 

n   OQO1 

7.  .0004128. 

19.  2143}  f. 

20.  22.901. 

9.  .8331. 

2.  .80448. 

20.  4101. 

27.  895i. 

70.  .5831. 

5.  .0000363. 

21.  13291. 

22.  A,  $648; 

77.  .714285f. 

4.  $43.216^ 

22.  1428ft. 

B,  $1080. 

72.  .444f  . 

$182.2875. 

25.  576ft. 
24.  1941|. 
25.  1460A. 

23.  $4360T7A. 
24.  336.  ' 

25.  $2475. 

75.  16.625. 
14.  27.92307644. 
75.  36.95831. 

5.  45.77125; 
55.02291f. 
6.  .273735; 

26.  923ft. 

26.  $2621. 

Art.  259. 

1.3136. 

27.  472£. 

27.  944. 

•1   1        1  /   1 

7.  3.39924; 

25.  10194,. 

28.  $964,;  361. 

7.  ^    14'  t- 

/ft   o        -/£*   1 

.409652. 

29.  810f. 

29.  Horse,  $705; 

2.  f    75.  1. 

6>   S       f/*   5 

5.  .00540625; 

50.  396f. 

Carriage, 

5.  I    76.  T%. 

.8455375. 

Art.  236. 

$440f. 

50.  $20031. 

5.  |    75.  !? 

9.  11.208704; 
.0100672.  - 

7.  Ij.   27.  26£. 
2.  20.    22.  71. 

57.  $157.67. 
52.  501. 

7!  5hr.  2o!  X! 

o   16     $)  1    3 

70.  5.7054831; 
34.01345|. 

5.  371.   ^-  15- 

55.  $9.46. 

°-  T2T*   *#•  ^6U* 

Q      9      O®   5 

77.  28.6480831; 

4.  63.   24.  21. 
5.  1171.  25.  10. 

34.  123TV  gal. 
55.  Widow, 

Y/l   64     j^^   1 

21.984375. 
72.  .288;  44.0928. 

5.  11.   26.  22f 
9.  i*.   27.  121 

$2876.12; 
Each  child, 

S:  J:  S:  MA. 

75.  93.056831; 
4.02031£. 

10.  11.    25.  16f 

$1438.06. 

¥'  -  1  2         5' 

14.  115.6666f; 

77.  6|.    29.  18f 

56.  14031. 

«  ?T¥F* 

500.40291. 

72.  9.    50.  21g 

57.  $192. 

•^5.  byg-^. 

75.  .51153; 

75.  llff.  57.  15. 

55.  $4600. 

Art.  262. 

3.85331. 

14.  9.    52.  91. 
75.  6ft.   55.  13. 
76.  2yg-.   54«  46. 
77.  21.    55.  13. 
75.  If.    56.  23. 
19.  ft.    57.  331. 

39.  Lost  $0.381. 
40.  20191  f; 
24348|. 
41.  31963f; 
185517¥V 
42.  6224f; 

1.  492.319787. 
2.  7462.31526. 
3.  476.338U807. 
4.  2.6591587. 
5.  9710.27879. 
6.  1.83586255. 

76.  82.0166f; 
1061.1796|. 
77.  576;  432;  216; 
345.6. 

75.  170845.86. 

Art.  271. 

7.  1764.06. 

39.  131. 

43.  19479|  ; 

1.  .048. 

40.  ft;  A;5H- 

44.  $198.Ji. 
45.  19744. 

9.  215.2741JV 
70.  21.9026730|. 

2.  250. 
3.  104;  8.625. 
4.  1.914;  2.82. 

43.  2;  l|. 

46.  $84.24. 

Art.  265. 

5.  .875;  100.8. 

44.  24-;  2. 

47.  27.80. 

1.  3.9803. 

6.  481.5;  385.2. 

45.  6351;  1- 

48.  76.66. 

2.  .26971. 

7.  4.25;  6.2. 

49.  320  rods. 

3.  8999.1. 

5.  15.24706; 

Art.  23^. 

50.  6  days. 

4.  .4648. 

2.25. 

7.  ^. 

57.  Gained  2  cts. 

5.  16.6736. 

9.  .49;  82.6875. 

2.  -ff. 

52.  $629.30. 

6.  .010102. 

70.  .5694;  39. 

^  —  -V—  . 

55.  $5487.98. 

7.  $86.17. 

77.  18.66; 

4.  1718|. 

54.  110  bu. 

5.  2.126155. 

10.30152  +  . 

5.  1931|.. 

55.  $35.46. 

9.  1.728-J. 

12.  2722.02;  42. 

6.  862ft. 

56.  $136.99. 

70.  $121.141. 

75.  86.40;  69.12; 

7.  31. 

57.  $115.30. 

77.  $1727.93^. 

51.84;  138.24; 

5.  3023564,. 

55.  $316.74. 

72.  .924f. 

25.92. 

350 


A NS WERS . 


[Art.  271. 


14.  1800. 

77.  TV 

77.  2934|  sq.  yd. 

Art.  292. 

15.  3720. 

7/2.  176.27$. 

18.  3200  A. 

/-Pi 

70'.  12. 

13.  2. 

19.  354  da. 

'         3"S  i"?  * 

£  |  oz. 

77.  2.525. 

74.  576. 

20.  2160  cu.  ft. 

5.      A    HI 

18.  293.040015. 

75.  5. 

21.  42885d. 

"     9  <>  -1*1* 
/          3       HI. 

19.  .03|;  .12$. 

16.  3i 

22.  982  da. 

Tr"       320     ***• 

SO.  .16*  ;  .40. 

77.  331.2. 

23.  27782  Ibs. 

0".  £2.° 

21.  .06*;  .06. 

18.  $56.16. 

24.  7583d. 

•    ^40* 

7.  -1  v. 

£2.  .021;  .03i. 

79.  $1575. 

8   XT 

«5.  .02^;  .oof 

24.  .14?;  .41  1. 

20.  $108.99. 
21.  $143.06. 

^r«.  2S5. 
7.  £35  6s.  3d. 

*  ¥tf  *  • 

9.  J-r  bu. 
70.  £|. 

#5.  $361.60;  $452. 

23.  $42.07; 

&  75cd.83cu.  ft. 

26.  $10800; 

$61.91. 

3.  117  bu.  2  pk. 

Art.  /><«/*. 

$49.85. 

24.  $16.84; 

7qt. 

7.  £.04375. 

£7.  $36;  $0.82. 

$11.17. 

4.  1227  gal.  1  pt. 

£.  .6  T. 

28.  $21600; 

25.  $28.98; 

5.  27  yd.  1ft.  3  in. 

5.  £.925. 

$10285.71. 

$32.30. 

6.  7  mo.  6  da. 

4.  £14.7875. 

29.  $200; 

#6.  $108.22; 

7.  2  mi.  235  rd. 

5.  .90625  cd. 

$153.125. 

$47.41. 

5.  37  wk.  6  da,  15 

6.  £247.7375. 

30.  $256000; 

27.  $67.76; 

hr. 

7.  .4  A. 

$3555.56. 

$79.20. 

9.  2  hr.  38m.  57s. 

8.  £27.525. 

31.  13.569; 

£&  $25.09; 

70.  173  yd.  11  in. 

9.  $166.56. 

1037.647. 

$62.56. 

77.  £55  15s.  7d. 

70.  $004.02. 

32.  4.524;  4.026. 

29.  $33107.12. 

12.  329  vd. 

77.  15s.  4.5d. 

33.  1.403;  6.247. 

30.  B,  $10542.48; 

13.  8  T.  14  cwt. 

Art.  296. 

34.  1.728;  l.«9. 

C,  D,  E,  each 

16  Ib. 

55.  9.681;  414.881. 
36.  142.105; 

$7028.32. 
31.  $31444.87. 

14.  50  yd.  9  in. 
75.  73  bu.  2  pk. 

1.  £85  7s.  lOd. 
2.  69  T.  14  cwt. 

C*(*    11 

98.743. 

32.  203f£A.; 

3  qt.  1  pt. 

oo  Ib. 

Art.  274=. 

1.  $6.        2.  $34. 
3.  $48.13. 
4.  $92.96. 

B,  36f£  A.; 

$4575.66.    ' 

33.  7040  ft. 
34.  Wife,  $48500; 

16.  £34  17s.  6d. 
77.  33  A.  36  sq.  rd. 
18.  4  cwt.  63  Ib. 
10  oz. 
19.  68  cd.  12  cu.  ft. 

3.  £460  7s.  6d. 
4.  12  y.  2  m.  16  d. 
5.  15  cd.  48  cu.  ft. 
6.  26  h.  46  m. 
7.  37  bu.  5  qt. 

o      ory        1       <    " 

5.  $312. 

6.  $5.44. 
7.  $114.58. 
8.  $102.24. 
9.  $280.64. 
70.  $104.1)4. 

Son,  $36375; 
Daughter, 
$24250; 
Total,  $145500. 
35.  $1065. 
36.  $4.63. 

20.  £20  2s.  5d. 

Art.  288. 
1.  135,1. 
2.  13s.  9d. 
3.  1980  ft. 

8.  27  yd.  4  m. 
9.  10  Ib.  2  pwt. 
20  gr. 
70.  124  gal.  2  qt. 
77.  £205  12s.  5d. 
12.  £546  Is.  7d. 

77.  $16.32. 

37.  1100  bu. 

4.  496  Ibs. 

Art.  298. 

12.  $12.40. 

13.  $15.75. 

Art.  282. 

5.  15s.  5d. 
6.  13  pts. 

1.  £88  4s.  2d. 
2.  £21  4s.  4d. 

74.  $38.75. 

1.  2326d. 

7.  4785  ft, 

3.  £3  14s.  3d. 

75.  $242.10. 

2.  220  gills. 

8.  87  Ibs.  8  oz. 

4.  £23  13s.  lOd. 

70'.  $16.50. 

3.  108404  f. 

9.  5^  pts. 

5.  4  y.  3  m.  2  d. 

77.  $198.40. 

4.  13265  Ib. 

70.  104  cu.  ft. 

6.  4  y.  4  m.  22  d. 

Art.  275. 

5.  256  days. 
6.  18211d. 

Art.  29O. 

7.  6  m.  10  d. 
8.  2  y.  6  m.  24  d. 

1.  .1651386. 

7.  235923  f. 

7.  150d. 

2.  289.36241*. 

8.  1566  qt. 

2.  17s.  6d. 

Art.  3OO. 

3.  .8125. 

9.  14948  Ibs. 

5.  153d. 

1.  £122  14s.  8d.  ; 

4.  158.916|. 

70.  4480  pwt. 

4.  13s.  8d. 

£157  16s.  ; 

5.  $117.16*. 

77.  27005  ft. 

5.  2  y.  4  m. 

£192  17s.  4d.  : 

6.  61.875. 

12.  2407680  ft. 

6.  £16  9s.  4d. 

£263. 

7.  3500. 

13.  572  pts. 

7.  £205  6s.  9d. 

2.  34  cd.  16cu.  ft, 

8.  .04. 

14.  Ill  pts. 

8.  2  y.  5  m. 

3.  £28  2s.  6d. 

,9.  |£. 

75.  69816  in. 

9.  £i5  6s.  9d. 

4.  3  Ib.  3  oz.  12 

70.  863.68964. 

16.  7895£  sq.  ft. 

70.  3  m.  3  d. 

pwt. 

Art.  271.] 


ANSWERS. 


351 


5.  16  h.  23  m. 

Art.  311. 

7.  900  sq.  ft. 

Art.  325. 

40s.; 
19  h.  40  m. 
24s.; 

1.  7  mo.  18  da. 
2.  8  mo.  18  da. 

8.  1386  sq.  ft,  ; 
154  sq.  yd. 
9.  960  sq.  ft. 

1.  123  cd. 
2.  302400  cu.  in. 
3.  1384  cu.  yd. 

24  h.  35  m. 

U(]» 

10.  55. 

4.  67  cds.  119 

30s.; 

Cld. 

11.  588  a. 

cu.  ft. 

29  h.  30  m. 
36s. 
6.  55  da.  13  h. 
7.  £20  12s.  6d. 
8.  £6  7s.  lOd.  ; 
£7  5s.  9d. 
9.  £3  9s.  3d.  ; 
£5  10s.  9d.  ; 
£8  6s.  Id.  ; 
£9  13s.  lOd. 

Art.  3O2. 

1.  £14s.9d.; 
£1  10s.  3d.  ; 
8s.  3d. 
2.  192yd.; 

4.  2  yr.  5  mo. 

7  da. 
5.  10  mo.  8  da. 
6.  9  mo.  12  da. 
7.  9  mo.  12  da. 
8.  4  mo.  15  da. 
1.  230  days. 
2.  263  days. 
3.  618  days. 
4.  888  days. 
5.  312  days. 
6.  286  days. 
7.  286  days. 
8.  138  days. 
9.  6574  days. 
11.  Dec.  6. 
12.  June  22. 

12.  132.183  a. 
13.  1.00352  a. 

14.  23f  sq.  vd. 
15.  17640  sq.  ft. 
16.  24  ft. 
17.  $7200. 
18.  990  ft. 
19.  $  sq.  ft.  ;  £  sq. 
ft.  ;  |  sq.  ft. 
20.  22-|  sq.  yd. 
21.  $440. 
22.  300  sq.  ft, 
23.  8000  sq.  yd. 
24.  19800  sq.  ft. 
25.  -fa  sq.  yd.  ; 
imr  sq.  yd; 

5.  560*cu.'  yd. 
6.  $72. 
7.  $26.95. 
5.  8  ft. 
9.  3f  cd. 
10.  135  cu.  ft. 
11.  233280  cu.  in. 
12.  168  tons. 
13.  $70; 
£21  17s.  6d. 
14.  213500  cu.  ft. 
15.  21f  perches. 
16.  27. 
17.  250  cu.  ft. 
18.  1260  cu.  ft.; 
26460; 

324yd.; 

352  yd. 

Art.  315. 

26.  108. 
£7.  10368. 

$238.14. 
19.  34944. 
20.  2112  cu.  ft. 

3.  44  yd. 

/   ry°  A  '  Q  O  1  f  /  • 

1.  93  ft.  1  in. 

28.  \\\  sq.  ft. 

21.  2240;  1120. 

ty.   (   0   O^T|-   , 

5°  41'  14"  ; 

2.  22£  in. 
3.  89760  ft.  ; 

£3.  1512. 
30.  17820. 

22.  3686.4  cu.  ft.; 
36  bbl.  ; 

4°  44'  21f"  ; 
3°  9'  34f". 
5.  40a.514sq.rd.; 

577£  ft. 
4.  1£  in. 
5.  9  mi.  100  rd. 

<?!.  187200. 
32.  $1040. 
«?<?.  $153600. 

28.8  cu.  yd, 
23.  102375. 

32  a.  41  sq.  rd.  : 
24a.180sq.rd.; 

20  a.  25|  sq.  rd. 
6.  9s.  lid.; 
£5  19s.; 
£9  18s.  4d.  ; 
£18  6s.  lid. 
7.  £34  10s.  ; 
£21  11s.  3d.  ; 

1  yd.  2  ft. 
6.  6514  in. 
7.  4620  ft. 
8.  66  ch.  20  ft.  ; 
52512  in. 
9.  99  ch.  ; 
6534  ft. 
10.  2478.96  ft.  ; 
150.24  rd. 

#4.  $1600;  32  a. 
35.  612  sq.  ft. 
36.  122  sq.  in. 
37.  62  sq.  yd.  ; 
26f  sq.  yd. 
38.  128  sq.  yd. 
39.  772  sq.  ft. 
40.  3631H-  sq.  ft. 
41.  4  yd. 

Art.  328. 

1.  12  sq.  ft. 
2.  48  sq.  ft. 
3.  480  sq.  ft.  ; 
$6.72. 
4.  5700  ft. 
5.  12  ft. 
6\  9|  ft,  7.  8  ft. 
5.  21  ft. 

£17  5s.; 

11.  6i  in.  ;  8|  in.  ; 

^.  24;  28.8. 

9.  16  ft. 

£9  ils.  8d.; 

Hi  in.  " 

43.  576;  900;  720. 

10.  33  1  ft. 

£5  15s. 

12.  287  fathoms; 

44.  51  f;  102$; 

ll!  24  ft. 

Art.  3O8. 

522f  fathoms. 
13.  9  mi.  45  rd.  4 

180. 
45.  24000;  33600; 

12.  54  ft. 
15.  480  sq.  ft. 

1.  26920  min. 

yd.  1  ft.  6  in. 

38400. 

14.  12096ft.; 

2.  232  da. 
3.  2  yr.  4  mo.  15 

4.  $0.69. 

14.  117.46  ch. 
15.  $708.50. 
16.  18480  posts; 
739200  ft. 

46.  750;  930. 
47.  f  sq.  yd. 
48.  $64. 
^9.  5  widths  and 

$181.44. 
15.  240  ft. 
16.  1280  ft. 
17.  8800  ft. 

5.  2  yr.  2  mo. 

4  in.  ;  35  yd. 

18.  661  posts. 

16  da. 

Art.  321. 

50.  25  yd. 

19.  240  ft, 

6.  51  hr.  8  min. 
7.  3;  7;  3. 

1.  175  a.  140  sq. 
rd. 

51  18;  10yd. 
52.  4  sq.  yd.  ; 

20.  184  ft, 
£1.  21  ft. 

8.  8  h.  22  m. 
50s. 

2.  130680  sq.  ft. 
3.  22405  sq.  yd. 

36  sq.  ft. 
53.  15  rolls. 

Art.  331. 

9.  1  d.  3  h.  46  m. 

4.  46  a.  50  sq.  rd. 

54.  16. 

1.  71  pt. 

40  s. 

5.  34,511,360  a. 

55.  2;  5. 

2.  109  gal.  1  qt. 

17. 

6.  2500  sq.  ft. 

56.  6  rolls. 

Ipt. 

352 


ANS  WERS. 


[Art.  331. 


3.  28  gal.  3  qt. 

24.  139^  bu. 

2.  $2.94. 

28.  11  hr.  20  min.. 

1  pt. 
4.  3174  bbl. 

25.  162±*bu. 
26.  18^  bu. 

3.  288;  960. 
4.  792. 

A.M. 

29.  4  A.  M.  Tues- 

19 gal. 

27.  86JL!  bu. 

5.  22$*. 

day. 

5.  576  gal. 

25.  lOS1^  bu. 

6.  $5.61. 

30.  122°  26'  45" 

6.  133.68+  cu.ft 

29.  324^0  bu. 

7.  $345.60. 

W. 

7.  1300  gal. 

SO.  221J-£  bu. 

5.  $9.18. 

31.  87°  37'  45"  W. 

9.  538.56  gal. 

57.  416^  bu. 

9.  50. 

10.  2393.6  gal. 

32.  190^  bu. 

Art.  359. 

77.  38297.6  gal. 

55.  279^  bu. 

Art.  347. 

IS.  1504  gal. 
14.  56400  gal. 
75.  169200  gal. 
16.  f.  3  2901. 
77.  7  gal.  3  qt. 
18.  72. 

35.  $153.54. 

36.  $142.92. 
37.  $167.13. 
38.  $233.40. 
39.  $237.45. 
40.  $119.02. 

1.  151°  50*  1". 
2.  134°  53'  59". 
3.  T  28'  46". 
4.  65°  56'  20"  ; 
197°  49'. 

5f\°  At  /IT//  . 

'  .38364  Km. 
4.  .001  7516  Km.; 
1.7516  m. 

5.  8742.57  m. 
6.  119  Km. 

7    4bQft  70 

Art.  333. 

41.  $295.38. 
42.  $200.67. 

,O     <±    4  /     , 

6°  20'  58f". 
6.  11°  58'  49"; 

/.    .Jpt/O.  i  U. 

8.  2306.8  m. 
5.  910  m. 

1.  187  pt. 
2.  156  bu.  1  pk. 

43.  $94.75. 

44-  $84.29. 

11°  13'  53TV" 
7.  277°  46'  40". 

10.  650. 

77.  $2400. 

3.  $25.60. 

4.  16128  1.5  cu.  in. 
5.  150  bu. 

Art.  342. 

1.  £135  Is.  lOd. 

8.  159400". 
9.  180°. 

12.  10240  m. 
75.  24  hr. 

^    6.  321.4,  ex.  ; 

QO  A      f\  vkv\ 

2.  £6  15s.  8d. 
3.  £42  10s. 

Art.  353. 

Art.  362. 

O-4U,   app. 

7.  160  bu.,  app. 
8.  235.2  bu.,  app. 
9.  640  bu.,  app. 

4.  £34  17s.  lid. 
5.  3080  a. 
6.  £18  8s.  lid. 
7.  £901  5s. 

1.  73°  54'  25". 
2.  73°  23'  52". 
3.  34°  49'. 
4.  178°  34'  17". 

2.  83.09  sq.  m. 
3.  4700  sq.  m. 
4.  602500  ca. 
5.  256  sq.  m. 

8.  £49  7s. 

5.  88°  33'  45". 

6.  400. 

-4r«.  34O. 

9.  £1  16s.  2d.; 
£2  6s.  6d.  ; 

0.  149°  14'  13". 
7.  4  hr.  56  min. 

7.  166.4  H. 

5.  $12.44. 

1.  1  Ib.  8  oz. 

10s.  6d. 

£  sec. 

9.  40. 

16  pwt.  16  gr. 
2.  15744  gr. 

10.  39  A;  51  f|. 
77.  £10  10s. 

5.  54  min.  30f 
sec. 

70.  96  H. 

3.  859  oz.  7  pwt. 

13.  £.808. 

9.  11  hr.  2  min. 

Art.  365. 

12  gr.; 
1100  oz. 

14.  £.921. 
75.  £.533. 

59T8y  sec. 
70.  50  min.  11T7F 

2.  17.218027 
cu  in 

4.  53  oz.  15  pwt. 

16.  £.867. 

sec. 

?    28  on    in 

5.  803  oz.  15  pwt. 

17.  £.363. 

77.  5  hr.  4  min. 

O*    /CO   Cll.    ill* 

A   3.3  cu.  in. 

6.  $42.98. 
7.  If  oz. 
8.  $68.75. 
9.  1,546,875  oz.  ; 

18.  £.696. 
20.  8s.  6d.  ;  12s.  9d. 
21.  16s.  4d.;  4s. 
lid. 

20f  sec. 
7^.  11  hr.  58  min. 
IHfsec. 
75.  2  hr. 

5.  4.81208  cu.  m. 
6.  2051.28  cu.  m. 
7.  $109.69. 
5.  82.5  sters* 

171875  oz. 

22.  5s.  Id.  ;  3s.  8d. 

14.  1  hr. 

4.Q  O  rr» 

10.  10480  gr. 

25.  7s.  6d.;  9s.  lid. 

75.  3  hr. 

rcO.iO  111. 

77.  1152;  1400. 

24.  £158  6s.  8d.  ; 

16.  1  hr. 

A  *t     ¥fi~ 

12.  51  Ib.  12  oz. 

£79  3s.  4d.  ; 

77.  2  hr. 

•         t  • 

13.  112000  gr. 

£63  6s.  8d.  ; 

18.  0  hr. 

1.  1000  1. 

14.  85  Ib.  16  pwt. 

£39  11s.  8d. 

75.  15  min.  46  sec. 

2.  $1.28. 

16  gr. 

25.  16s.  8d.  ; 

20.  40  sec. 

5.  72  HI. 

75.  192  oz.  av. 

£1:   13s.  4d. 

21.  9  min.  47  sec. 

4.  168  HI. 

16.  $16.31. 

26.  $1100.25; 

22.  \  min.  1  sec. 

5.  6351.; 

77.  $10.06. 

$683.91. 

23.  9  min.  29  sec. 

8375  1. 

18.  $321.66. 

27.  £204  18s.  4d.  ; 

24.  20  min.  19f 

6.  $1498. 

20.  $91.46. 

£332  19s.  lOd. 

sec. 

7.  160  HI. 

21.  $94.76. 
22.  $73.92. 

Art.  345. 

25.  154°  8'  30"  W. 
26.  5  P.  M. 

5.  180  bags. 
9.  10080  Dl. 

23.  $70.73. 

1.  $28.80. 

27.  19°  31'. 

70.  $14. 

Art.  369.] 


A  NS  WE R  S. 


353 


Art.  369. 

21.  16500  ft. 

11.  $60.10. 

14.  $3500. 

1.  1,  000,000  g.; 
1000  e- 

00.  $327.78. 
23.  $1634.09. 

12.  $33.17. 

13.  $52.55. 

15.  $720. 
16.  $59.38. 

AVJW     g. 

0.  16.816  T. 

24.  $808.65. 

!,£.  $1031.97. 

17.  $2500. 

5.  80. 

25.  £19.8375. 

15.  $155.36. 

IS.  $1944. 

^.  30. 

26.  $1100.25. 

16.  $23.75. 

19.  2500. 

5    750  P- 

07.  £15  16s.  Id. 

17.  $112.68. 

00.  2400. 

t>.     4<JU    g. 

6.  $13.608. 

28.  £10  10s. 

18.  $143.75 

01.  $361.60. 

7.  62 

29.  55°  48'. 

$2731.25. 

22.  $3240. 

8.  9300  Kg. 

30.  165000  1. 
51.  $0.526. 

19.  $16121. 
20.  $225. 

05.  $1502.40. 
24.  $69.12. 

/<*»/    ^y/ 

50.  $0.055. 

21.  $823.82. 

05.  495  ft. 

j."  I  /   »  •     t>  /  Jt  • 

00.  $1612.50; 

26.  £7200. 

,1.  246.06  yd.  ; 

Art.  389. 

$2700.40. 

07.  $196. 

8858.25  in. 

-t     db~t  o  nn  . 

23.  $2325.38. 

28.  $15360. 

0.  9.6558  Km. 

1.  $18.99; 

24.  $84. 

09.  $91500. 

5.  259.008  H.  ; 

$20.34  ; 

26.  $0.94. 

50.  $65500. 

25900.8  A. 

$16.55. 

07.  $6.56. 

31.  $564. 

4.  32808.3  ft.  ; 

0.  $2.06. 

28.  $14.71. 

50.  400. 

6.2137  mi. 

5.  $12.16. 

29.  $3.96. 

33.  $930. 

5.  828.04776  Ib. 

4.  $13.50. 

30.  $79.13. 

5£  $444. 

6.  204.12. 
7.  26.73  grams; 
27.216  grams. 
8.  1762  HI. 
9.  668.9375  cu.  m. 

5.  $5.42. 
6.  $12.16. 
7.  $16.39. 

5.  $32.89. 
9.  $3.44. 

51.  $53.83. 
50.  $1476.40. 

Art.  4O4. 

55.  $324. 
56.  $6210. 
57.  $456.80. 
38.  $505. 
39.  1425  boxes. 

10.  1308  cu.  yd. 

10.  $40.25. 

1.  432. 

40.  $59062.50. 

11.  6540.48  1. 
12.  291.824  sq.  yd. 
2626.416  sq.  ft. 

11.  $54.07. 
12.  $76.48. 
15.  $245.80. 

2.  868. 
5.  1604.5. 
£  1816. 

41.  $6464. 

13.  6237  g.; 
6.237  Kg. 

Art.  391. 

5.  $125. 
6.  $106. 

1.  .OH. 

2.  $179.80. 

7.  $162. 

'        QJ?' 

Art.  387. 

3.  $49.50. 

<?.  $1975. 

/'•  "osX 

1.  35AV 

4.  $193.45. 

5    $168 

9.  $243.60. 

10.  $487.10. 

•*•    •  V/~'T5* 

5.  3|%. 

0.  1135.94. 
5.  5  mo.  24  da.  ; 
178  da. 
4.B. 
5.  32  ft. 
6.  702  sq.  ft.  ; 
210  sq.  ft.  ; 
1320  sq.  ft. 
7.  $172.36. 

6.  $277.53. 
7.  $4. 
5.  $289.39. 
9.  $96.25. 
10.  $21.62. 
11.  $7.56. 
10.  $36.60. 
15.  $85  ;  $107. 

11.  $176.43. 
12.  $328.35. 
14.  $2434.79. 
15.  $184.38. 
16.  $64.94. 
17.  $4100. 
18.  $4500; 
$16.88. 

9.  lf%'. 
10.  33^%. 
11.  25%. 
10.  ¥3o%. 
15.  2200. 

8.  $127.49. 
9.  43200  sq.  rd. 

14.  $18.75. 
15.  $81;  $90. 

Art.   4O7. 

15.'  45%.' 
16.  44%. 

10.  480  sq.  ft. 

16.  $274.81. 

1.  8. 

17.  \%  ; 

11.  3110ft. 

2.  800. 

$42.50. 

12.  $128. 

Art.   4O2. 

3.  5400. 

18.  4^%. 

15.  628  ft. 

1.  51.84. 

4.  5600. 

19.  16%. 

14.  15.708  ft. 

0.  60. 

5.  9375. 

00.  56£%. 

15.  28.27  sq.  ft.  : 

3.  17.92. 

6.  22.8. 

01.  4£%. 

78.54  sq.  ft. 

4.  23.22. 

7.  1800. 

00.  18|%. 

16.  429  ft. 

5.  56.7. 

5.  800. 

05.  6|%. 

17.  35942  Ib. 

6.  $33.12. 

9.  2500. 

24.  \%. 

18.  474  qt. 

7.  $57.28. 

10.  $34.56. 

25.  9^-%. 

19.  487  ±4  bu.  ; 

8.  $644. 

11.  $4050. 

06.  45%. 

$237.81. 

9.  $411.95. 

10.  $900. 

07.  21if%. 

20.  $237.94. 

10.  $41.04. 

15.  $228.80. 

28.  20%. 

354 


ANS  WE R  S. 


[Art.  41O 


29<  $320; 

22.  $72. 

51.  $40. 

17.  $2554.75. 

$280. 

23.  23i%. 

50    $250. 

15.  $43500. 

30.  12|%. 

04.  150%. 

33    $50. 

19.  $1773.50. 

51.  20%. 

25.  57^. 

54    $15.36. 

20.  $8450.89. 

32.  f%, 

06.  12$. 

56    50%. 

01.  $2433.90. 

33.  |%. 

07.  10%. 

57      31y^%. 

22.  $7872.07. 

28.  4%. 

55   13}°ff%. 

23.  $2175.44. 

Art.  411. 

09.  $210. 

59    $1.26f; 

24.  $50000. 

1.  1392. 

30.  36?. 

304%. 

25.  $2751.14. 

0.  120. 

51.  96%. 

26.  $852.91. 

5.  125$. 

50.  Loss,  4%. 

Art.  419. 

27.  $185.40. 

4.  $1197. 

33.  38f%. 

1.  $44.10. 

28.  $1388.63. 

5.  275. 

54.  $432. 

0.  $926.12. 

29.  $988. 

6.  $7500. 

35.  $4.25,  each. 

3.  $72. 

30.  $5136. 

7.  $16.88. 

36.  12%. 

4.  $533.22. 

31.  $61200. 

5.  30^. 

57.  $6.40. 

5.  $135.73. 

50.  $123.31. 

9.  $157790.70. 

10.  $28546.56. 
11.  $5.57. 

38.  $7.50. 
59.  6|%  ;     $3.80. 
40.  $2532.96; 

6.  $118.73.         Y 
7.  $360.07.         K 
5.  $900.43;        ^ 

33.  $495.19. 
34.  $1234. 

35.  $875.52. 

12.  $1150. 

$3102.88; 

$855.41. 

36.  $900. 

15.  $164. 
14.  $12960. 

22^%. 
41.  $400,  loss. 

9.  $149.05. 
10.  $340.64; 

57.  $288.39. 
38.  $18909.18. 

15.  $9363.44. 

16.  2f%. 

Art.  417. 

$333.83; 
$337.23. 

Art.  441. 

17.   $6907.51. 
15.  $123450; 

1.  $78.55. 
2.  $31.50. 

11.  $1082.55. 
12.  $145. 

1.  $122.50. 
0.  $115.65. 

$1313.28. 

3.  $206.55. 

13.  $65.64. 

3.  $525.25. 

19.  $6058.95; 

4.  $76.58. 

14.  $54.71. 

4.  $702.5l! 

95%;    20%. 
20.  $1052.63; 

5.  $40.30. 

6.  $18.24. 

15.  $258.32; 
$245.40. 

5.  $917.'72.' 
6.  $214.31. 

$1422.37. 
21.  $1079.12. 

5.  23|%  ;  50|%  ; 
30%  ;  64%  ; 

16.  $74.31; 

$72.82. 

7.  $535.50. 
5.  $397.49. 

22.  7£%;    29%. 
24.  £11  10s.  5d. 
25.  £26  9s. 
06.  £18  15s. 

78|%;42f%; 
74f%. 
9.  $106. 
10.  $40.83. 

17.  $27.43. 
15.  $49.76. 
19.  $1692.46. 
00.  $93.36. 

9.  $106.58. 
10.  $3393.33. 
11.  $1059.71. 
12.  $470.63. 

07.  £1  12s.  5d. 
28.  £3  Os.  6d. 
29.  £3  14s.  lid. 

11.  200  yd. 
10.  $354.06. 

15.  $102.82. 

01.  $50.12. 

00.  $359.58. 
23.  $1563.87. 

13.  $18.81. 
14.  $1791.94. 

30.  £32. 
31.  £35  8s.  4d. 

14.  $420.26. 
15.  $535.42. 

24.  $155.40. 

Art.  445* 

16.  $147.67. 

Art.  427. 

1.  $20. 

Art.  414. 

17.  $26.25. 

1.  $21.60. 

2.  $21.82. 

1.  $40. 

15.  $168.93. 

0.  $200000. 

3.  $1.79. 

2.  $364. 

19.  $905.23; 

3.  $212.50. 

4.  $2.92. 

5.  $4726. 

$896.08. 

4.  $2256.25. 

5.  $47.11. 

7.  $3. 

20.  $450.96. 

5.  $2295. 

6.  $0.91. 

5.  $10179. 

01.  42f%. 

6.  2%. 

7.  $6.48. 

10.  $1048.20. 

22.  33|-%. 

7.  480. 

5.  $6.81. 

11.  72/'. 

23.  40|%. 

5.  $3099.37. 

9.  $1.75. 

12.  $2425. 

24.  100%. 

9.  $12.61. 

10.  $16.16. 

14.  16  #. 

25.  331%  ;  50%  ; 

10.  $81.75. 

11.  $6.91; 

15.  13ff  %. 

53^1%  ; 

11.  $2161.17  net 

$5.76. 

16.  12^%. 

4Q|^%  •  42f  %  ; 

amount. 

12.  $127.58; 

17.  $1775. 

30^o%. 

12.  $65.63. 

$148.85. 

15.  $608. 

26.  31|-%. 

13.  $869.60,  com. 

15.  $101.23; 

.19.  $11225. 

07.  H-g-% 

14.  $125. 

$88.57. 

100.  $520. 
01.  44^. 

05.  58ff%, 
30.  $2.50. 

15.  5%. 
16.  $5091. 

14.  $53.23; 

$37.70. 

Art.  445.] 


ANS  WE RS. 


355 


15.  $50.69; 

34.  $107.36; 

68.  $16.28;  $14.65. 

Art.  466. 

$33.79. 

$80.52. 

$16.57;  $14.92. 

1.  1%.  10.  $%. 

16.  $663.33; 

35.  $93.53; 

69.  $53;  $59.63. 

2.  h%.  11.  8$fo. 

$1105.56. 

$140.30. 

79.  $46; 

3.  Qfc.  12.  $%. 

17.  $33.25;  $24.94 

36.  $19.27; 

$48.30. 

4.  5%.  13.  \§\%. 

$33.98;  $25.48. 

$14.46. 

$47; 

5.  1%.  14.  30%. 

18.  $23.16;  $28.95. 

37.  $494.15; 

$49.35. 

6.  §\%.15.  8%. 

$23.44;  $29.30. 

§47.07. 

71.  $9.80; 

7.  Sfc.  16.  Qfc. 

19.  $92.67; 

18.64; 

$8.98. 

8.  Uf0.17.  1%. 

$185.33. 

37.29. 

$10;  $9.17. 

9.  4%. 

$93.33; 

39.  $434.60; 

72.  $10.27; 

$186.67. 

$437.58. 

$9.54. 

Art.  469. 

20.  $90.35; 

40.  $1393.29; 

$10.50; 

$150.58. 

$1446.05. 

$9.75. 

1.  6  m.  6  d. 

$92.37; 

41.  $623.85; 

73.  $18.13; 

2.  1  y.  4  m.  20  d. 

$153.95. 

$615.81. 

$7.93. 

3.  2  y.  2  m.  2  d. 

21.  $654.50; 

42.  $2267.51; 

$18.56; 

4.  2  m.  15  d. 

$872.67. 

$2249.67. 

$8.12. 

5.  6  m.  14  d. 

$656.83; 

43.  $6503.40; 

74.  $14.04; 

6.  1  y.  10  m.  22  d. 

$875.88. 

$6597.60. 

$7.41. 

7.  4  y.  9  m.  15  d. 

44.  $4416.93; 

$13.88; 

£  4  y.  8  m.  34  d. 

Art.  459. 

$4390.69. 

$7.32. 

9.  9  m.  20  d. 

1.  $8.64. 

45.  $389.94; 

75.  $51.33; 

10.  11  m.  21  d. 

2.  :  1:88. 

$387.12. 

$38.50; 

11.  1  y.  5  m.  15  d. 

3.  >3.48. 

46.  $1450.87; 

$51.91; 

12.  1  y.  5  m.  18  d. 

/  ^  fid. 

$1605.74. 

$38.93. 

13.  16  y.  8  m. 

ty.  $o.O'±. 

5.  $6.57. 

47.  $5092.50; 

76.  $10658.20. 

14.  28  y.  6  m.  26  d. 

6.  $6. 

$5032.71. 

77.  $1050. 

7!  $5^34. 

48.  $2. 

78.  $1556.66. 

Art.  472. 

8.  |X. 

49.  $10.66. 

79.  $27.84. 

1.  $12107.84. 

£».  $7.89. 
JO.  $0.85. 

50.  $53.44. 
51.  $14.19. 

80.  $28.93. 
81.  $1356.18. 

2.  $871.31. 
3.  $2241. 

11.  $10.63. 

52.  $0.50. 

82.  $1562.50. 

4.  $7719.16. 

12.  $11.73. 

53.  $33.11. 

83.  $86.07. 

5.  $1997.87. 

75.  $18.14. 
74.  $0.31. 

54.  $61.20. 

55.  $8.38. 

84.  $5643. 
85.  $601.39. 

6.  $3000. 
7.  $3228.33. 

75.  $3.73. 

56.  $11.33. 

8.  $29419.35. 

16.  $10.07. 
77.  $35.63. 

57.  $9.33. 
58.  $34.96;  $33.21. 

Art.  463. 

1.  $1.79. 

9.  $30612.25. 
10.  $31746.03. 

18.  $9.50. 
79.  $26.04. 

$35.65  ;  $33.87. 
59.  $14.27;  $18.73. 

2.  $116.47. 
3.  $11.54. 

11.  $11973.33. 
12.  $14370.69. 

00.  $46.67. 

21.  $16.28;  $13.57. 
22.  $64.76; 

$14.47;  $18.99. 
60.  $26.  73;  $19.09. 
$27.29;  $19.49. 

4.  $5.29. 
5.  $3.92. 
6.  $15.12. 

IS.  $3436.99. 
Art.  475. 

$32.38. 

61.  $193.96; 

7.  $5.75. 

1.  $1234. 

23.  $36.85;  $42.99. 

$113.14. 

8.  $6.16. 

2.  $5280. 

24.  $24.  70;  $20.58. 

$195.15; 

9.  $42.18. 

3.  $3456. 

25.  $56.68;  $75.57. 

$113.83. 

10.  $14.83. 

4.  $375.60. 

26.  $180.10; 

62.  $309.07; 

11.  $4.44. 

5.  $12375. 

$120.07. 

$347.70. 

12.  $10.44. 

6.  $1728. 

27.  $11.43;  $17.  14. 

$310.14; 

13.  $39.35. 

7.  $723,01. 

28.  $39.45;  $46.03. 

$348.90. 

14.  $246.89. 

8.  $879.54. 

29.  $19.79;  $23.09. 

63.  $57.27;  $81.  81. 

15.  $58.97. 

9.  $1511.67. 

30.  $2.85;  $2.37. 

$57.82;  $82.60. 

16.  $27.74. 

10.  $2309.28. 

31.  $13.75;  $16.04. 

64.  $7.59;  $10.12. 

17.  $41.64. 

11.  $3770.52. 

32.  $106.66; 

65.  $25.95;  $11.53. 

18.  $7.58. 

12.  $5307.72. 

$142.22. 

66.  $111;  $27.75. 

19.  $2.14. 

13.  $1642.31. 

S3.  $137.72; 

67.  $76.50;  $60.56. 

20.  $30.21. 

14.  $2138.94. 

$114.77. 

$77.40;  $61.28. 

21.  $5. 

15.  $5063.11. 

356                                                    ANSW&KS.                                        [Art.  475. 

16.  $2863.86. 

3.  $83.26. 

9.  Sept.  1  ; 

39.  $1523.25. 

17.  $3590.09. 

4.  $1909.63; 

$4430,  or 

40.  $3081.09. 

$2104.72. 

$4430.96. 

Art.  479. 

5.  $1211. 

10.  Sept.  28; 

Art.  505. 

6.  $220.80; 

$8204.29,  or 

1.  $678.54. 

'  $107!  14.' 
2.  $438.60; 

$268.51. 

7.  $8583.80. 
A  $1811.44; 

$8204.78. 
11.  June  29  ; 
$4276.08,  or 

2.  $242.17; 
$148.16. 
3.  $1102.69; 

3.  $547.95; 
$52.05. 

$2564.94. 
9.  $2794.32; 

$7798.54. 

$4276.73. 
12.  Nov.  30; 
$4768.85,  or 

$1184.37. 
4.  $1327.21; 
$1410.94. 

<  'Qfy  Qry    ' 

10.  $993.03. 

$4770.44. 

5.  $835.74; 

5.  $283.35; 
$41.65. 
6.  $161.64; 

11.  $4445.17. 
12.  $2450.13. 

13.  $747.27. 
14.  $4172.57. 

13.  May  6; 

$8899.50,  or 
$8900.88. 
14.  Jan.  15; 

$924.38. 
6.  $898.88. 
7.  $3073; 
$3363.56. 

7.  $595.39; 
$204.61. 
8.  $641.79; 
$258.21. 
9.  $0.68. 
10.  $3629.03. 
11.  $204.29. 

15.  $405.34. 
16.  $13363.84. 
17.  $4659.94. 
18.  $343.90. 

Art.  495. 

1.  $1022. 

$4909.58,  or 
$4910.82. 
15.  Feb.  2; 
$5936.20,  or 
$5937.07. 
16.  $5949. 
17.  Oct.  5; 
$4946.67. 

8.  $517.82; 
$716.62. 
9.  $3260.23; 
$4539.19. 

Art.  5O9. 

2.  $440; 
$447.16. 

Art.  480. 

1.  $42.32. 
2.  $1843.93. 

2.  $911.04; 
$919.21; 
$917.46. 
3.  Mar.  7,  1890; 

18.  Apr.  4; 
$3710. 
19.  Aug.  6; 

$6882.75. 

3.  $223.31; 

$214.37. 
4.  $651.97; 
$753.30. 

4.  ly.'  10m.  28  d. 

$6022.10. 
4.  $431.10; 

20.  July  4; 

$8909.75. 

Art.  513. 

5.  $1722.02. 
7.  Uf#. 
8.  Latter  \\% 
better. 

'  $42485. 
6.  Aug.  21,  1887. 
7.  Monday. 
8.  Saturday  ; 

21.  Dec.  21  ; 
$4838.61. 
22.  Sept.  29  ; 
$4451.25. 

2.  $262.24; 
$161.49. 
3.  $1108.57; 
$1192.53. 

9.  Oct.  3. 

Wednesday. 

23.  Dec.  5; 

4.  $1324.75; 

10.  1%. 
11.  $19230.77. 

9.  Oct.  23. 

$8870.37. 
24.  Dec.  8; 

$1406.47. 

5.  $833.87; 

12.  $4298.04; 

$2970.75. 

$920.94. 

$4342.65. 

Art.  5OO. 

25.  June  24; 

6.  $897.77. 

13.  June  11,1874. 
14.  $129; 
$131.50; 
$129.70. 
15.  $650. 
16.  $1483.98. 
17.  $89.17. 
18.  $606.60. 
19.  £199  3s.  8d. 

2.  $7937.33,  or 
$7938.19. 
3.  Apr.  27; 
$1181.40,  or 
$1181.65. 
4.  Aug.  21; 
$5196.40,  or 
$5197.55. 
5.  Nov.  2* 

$98)2.71. 
26.  Oct.  11; 
-  $5894.25. 
27.  May  13; 
$5897.87. 
28.  Sept.  6; 
$8603.70. 
29.  $7837.33. 
30.  $8808. 

7.  $3067.14; 
$3347.31. 
8.  $549.89; 
$764.05. 
9.  $3260.51; 
$4594.82. 

Art.  523. 

1.  12. 

20.  £8  10s. 
#1.  £5  8s.  4d. 
22.  £9  5s.  lOd. 
23.  £2  12s.  lid. 
24.  £3  17s.  4d. 
.95.  £10  9s.  8d. 

^ir«.  485. 

'  $2524.16,  or 
$2524.65. 
6.  Oct.  7; 
$3664.71,  or 
$3665.96. 
7.  Nov.  18; 
$6395.55,  or 
$6395.96. 

31.  §\%. 
32.  $15.87. 

33.  $3012.09. 
84.  Aug.  30; 
$3737.21. 
35.  May  19; 
$1641.17. 

3.  160. 

4.  $36. 
5.  $50. 
6.  240  Ib. 
7.  $111. 
8.  $21.875. 
9.  $55.50. 
10.  85|yd. 

1.  $526.44; 
$506.48. 

8.  Sept.  16; 
$8135.73,  or 

36.  Apr.  15; 

$882.22. 

11.  396  ft. 
12.  £25  11s.  9d. 

£.  $45.18;  $37.37. 

$8139.01. 

38.  $1523.62. 

13.  $960. 

Art.  523.] 


A  N  S  WE  R  S. 


35? 


14.  $2410.71. 

7.  $22.50. 

Art.  567. 

62.  147. 

15.  A,  $1875.90; 

8.  $56.88. 

1.  $1083.94. 

63.  13950.75  fr. 

B,  $1598.40. 
16.  15.9883:1. 

9.  $71.43. 
10.  $57.60. 

2.  $1583.40. 
4.  $2407.50. 

64.  £391  4s. 
65.  7343.75  fl. 

17.  $179.56. 
18.  $126. 

11.  $4880. 
12.  $28.56. 

5.  $3760.69. 
6.  $4050.03. 

66.  £1009  7s.  64 
67.  £512  3s.  2d. 

19.  $3.79. 
20.  $137.03. 

13.  $42. 
14.  $9000. 

7.  $409.34. 
8.  $2483.15. 

Art.  574. 

31.  $153600. 

15.  $9000. 

9.  $4076.72. 

1.  Oct.  10. 

?£  2355|  ft. 

16.  40%. 

10.  $3290.93. 

2.  May  11. 

23.  25.215  fr. 

17.  $16.20; 

11.  $1087.98. 

3.  Feb.  11. 

V4.  252f  f  yd. 
85.  20  hr.  45  min. 

$27. 
18.  $1440. 

12.  $4261.23. 
13.  $4593.93. 

4.  Aug.  4. 
5.  June  5,  1882. 

9  sec. 

15.  $61.25. 

14.  $2611.06. 

6.  June  20. 

26.  $11.10. 
27.  69  da. 

20.  $792. 
£Z.  M,  $1761.36; 

15.  $8495.46. 
16.  $2373.24. 

7.  July  18; 
$1694.90; 

28.  $3000. 

P,  $1409.09; 

18.  £1225  18s.  6d. 

$1686.16. 

29.  $912.23. 

T,  $880.68. 

19.  4831 

8.  Feb.  15,  1882. 

30.  47i  yd. 

22.  A,  $454.54; 

20.  £1864  6s.  4d. 

9.  Oct.  6,  1881  ; 

31.  4449TV  bu. 

C,  $568.18. 

21.  $1341.32. 

$2403.88  ; 

32.  $84.18. 

23.  2.767%. 

22.  $1162.79. 

$2367.55. 

55.  $0.73. 

'24.  $2627.78. 

23.  $965.02. 

10.  May  31  ; 

34.  $3.12. 

25.  $315.33. 

24.  $767.20. 

$2480.32; 

55.  $17.60. 

26.  $28.03. 

25.  $1631.38. 

$2492.72. 

36.  $22.67;  $25.33. 

27.  $155.70. 

26.  $1393.72. 

11.  July  14. 

37.  45  days. 

27.  $189.60. 

12.  Nov.  15. 

38.  $1.09$. 

59.  1492.26ft. 

Art.  553. 

28.  $1856.08. 
29.  $2023.10. 

13.  Sept.  5. 
15.  Oct.  26,  1882. 

46>.  18|  mo. 

30.  $1534.69. 

16.  Sept.  25  ; 

41.  $5328,  assets; 
$11100,  debts. 

1.  $8782.81. 
2.  $8395.94. 

31.  $2386.62. 
32.  $1688.75. 

$2425.16; 
$2437.33. 

Art.  526. 

1.  $105.63. 

3.  $5006.25. 
4.  $4358.59. 

5.  $8427.52. 
6.  $9922.37. 

55.  $1411.11. 
34.  80318.70  fr. 
35.  17972.04  fr. 

36.  5.18|. 

17.  July  8. 
18.  Mar.  21,  1882. 
19.  Feb.  13,  1882. 
20.  April  12,  1883. 

2.  6  hr. 
5.  126  A. 

7.  $5270.79. 

8.  $4287.11. 

38.  $1692. 
39.  $1967.24. 

Art.  582. 

£  $4665.60. 

9.  $3417.18. 

40.  $822.96. 

1.  Jan.  17,  1889. 

5.  4hr. 

10.  $2821.96. 

41.  $289.54. 

2.  May  25.  1888. 

6.  125. 

11.  $9898.30. 

42.  $2359.10. 

3.  Dec.  3,  1882. 

7.  I  day. 

12.  $8406.44. 

43.  $1543.24. 

4.  May  13,  1882. 

8.  4  days. 

U.  $1871.48. 

44-  $3867.75. 

5.  Sept.  6,  1882; 

9.  15  hr. 

15.  $2443.86. 

45.  $3102.24. 

$276.53. 

10.  9  days. 

16.  $3346.56. 

46.  $1547.25. 

6.  Feb.  15,  1883; 

11.  $64. 

17.  $2227.28. 

47.  $2061.40. 

$1053,23. 

7,?.  80  days. 

18.  $8144.96. 

48.  $2359.33. 

7.  Feb.  8,  1881. 

13.  $66.13. 

19.  $7373.16. 

49.  3467.73  marks. 

8.  Aug.  25,  1882, 

14.  $1493.33. 
15.  32  days. 

20.  $9222.61. 
21.  $6431.57. 

50.  13824  marks. 
51.  94|. 

Art.  586. 

16.  16320  Ib. 

22.  $9231.46. 

52.  $2905.05. 

1.  $2074.06; 

23.  $5283.96. 

53.  $4976.40. 

Aug.  28; 

Art.  544. 

24.  $23.29; 
$9340.10. 

54.  4562  guilders. 
55.  7128  guilders. 

Sept.  2; 
Sept.  2. 

1.  $93.75. 

25.  $2972.25. 

56.  40f  . 

2.  $5324.48; 

2.  $68.75. 

26.  $11834. 

57.  4.885. 

Nov.  28. 

3.  $354.50. 

27.  $7854.67. 

58.  $17366.99. 

3.  $2751.14; 

£  80^. 

28.  \%  premium. 

59.  $3.60. 

Sept.  27. 

5.  $3600. 

29.  $3420.05. 

60.  118.4;  117.1. 

4.  $12505.70; 

6.  $281.25. 

30.  $10118.89. 

61.  85.98. 

Apr.  26. 

358 


A  NS  WER  S. 


[Art.  586. 


5.  $4043.09; 

33.  $20.80. 

19.  $15000. 

11.  A,  $6470.24; 

Dec.  10. 

34.  $45,066,444.72 

20.  $1.1692. 

B,  $3235.12. 

35.  100. 

21.  1.65%. 

12.  $1510. 

Art.  594. 

56.  $66,000,000; 

22.  A,  $177.63; 

14.  E,  $2380.83; 

1.  $431.37. 

$14,000,000. 

B,  $305.42. 

F,  $3333.17; 

2.  $986.02. 

37.  80. 

23.  $5887. 

G,  $3809.33; 

3.  $3361.51. 

5&  8%;  6%;  51%. 

H,  $4761.67. 

4.  $1694.89. 

39.  $176;  33. 

Art.  628. 

15.  A,  $2692.68; 

5.  $518.53. 

40.  8%. 

1.  $9412; 

B,  $2468.29; 

6.  $44955.75. 

4&.  Latter  yV% 

$5647.20. 

C,  $1884.88; 

7.  $400.91. 

better." 

2.  $97.02. 

D,  $2154.15. 

8.  Rm.  9997.87. 

43.     7y\  %  . 

3.  $80.49. 

16.  A,  $375; 

9.  $276.54. 

^.  Chatham   $60 

^.  $816.25. 

B,  $318.75; 

10.  $1053.22. 

greater. 

5.  $85.40. 

C,  $225; 

11.  $1513.77; 

45.  166f. 

C.  $135.21. 

D,  $187.50; 

$32067.54; 

46.  160;  1331. 

7.  $723.45; 

E,  $131.25. 

$1704.29; 

.^7.  6|-%;  125;  621 

$748.80. 

17.  A,  $1533.46; 

$47288.32. 

48.  $2425. 

8.  $950; 

B,  $1922.54. 

12.  Rm.  3869.18. 

^5.  $7,725,574.22; 

$609.60. 

18.  C,  $819.97; 

13.  $52,23. 

41.48%. 

9.  $1267.50, 

D,  $745.43. 

14.  $856.19. 

50.  $754482; 

10.  $116.94. 

19.  A,  $540; 

15.  $622.42. 

75.6%. 

11.  $807.80. 

B,  $560; 

16.  $4016.22. 

5.Z.  $3,957,320; 

12.  $1004; 

C,  $600. 

$49,466,500. 

$794.25. 

20.  A,  $2311.63; 

Art.  611. 

52.  $519.27. 

13.  $639.03. 

B,  $3581.40; 

1.  $8750;  $56. 

53.  $1128.34. 

14.  $839.40. 

C,  $4106.97. 

2.  8%  ;  $200. 
3.  $2,500,000. 

54.  $1682.91. 
55.  $4030.29; 

15.  $1623.80. 
16.  $183.90. 

21.  J,  $1558.97; 
K,  $1385.75; 

4.  $20,000,000; 

$4030.29. 

17.  $338.24. 

L,  $1190.88. 

$2.000,000; 

56.  $475. 

18.  $11060. 

22.  A,  $1529.98; 

1%. 

57.  $1450. 

19.  $43.75. 

B,  $1185.74; 

5.  $500. 

58.  $1625. 

20.  $208.50. 

C,  $1070.98; 

6.  $61250. 

60.  124.59; 

21.  $439.88. 

D,  $940.30. 

7.  $36.745,000. 

•  128.80. 

•22.  $351. 

23.  $1750. 

8.  $60000. 

62.  4%. 

23.  $1959. 

24.  $11287.62. 

9.  $23325. 

24.  $57.05. 

25.  R,  $1925; 

10.  550  shares. 

Art.  617. 

25.  $37242; 

S,  $1425; 

11.  $240,000; 

1.  $7,690,418.82. 

$197.75. 

T,  $1125; 

$185,237.50. 

2.  $11,615,280. 

26.  $2483.60. 

U,  $925. 

12.  $23100. 

3.  31  mills; 

27.  $487.27. 

26.  X,  $500; 

13.  $36412.50. 

$38,666.37; 

28.  $156.30. 

Y,  $200; 

14.  $226675. 

$11,986.57. 

Z,  $700. 

15.  $17560. 

4.  Rate  5.8  mills  ; 

Art.  639. 

27.  65%. 

16.  $29043.75. 

$42.34. 

3.  A,  $1960; 

A,  $1235; 

17.  $68625: 

6.  $37.49. 

B,  $2960. 

B,  $3250; 

18.  $3277.50. 

7.  $57.85. 

4.  A,  Or.  $1623.  17; 

C,  $1950; 

19.  $16200.  20.  8. 

8.  $231.39. 

B,  Dr.  $164.71. 

D,  $3965. 

21.  $8230. 

9.  $60.48. 

5.  $833.33,  bro.  ; 

28.  55%; 

22.  $80800. 

10.  $597.85. 

$3366.67. 

$9020. 

23.  $775. 

11.  $104.98. 

6.  C,  $11431.88: 

29.  42%  ; 

24.  500  shares-. 

12.  $65.28. 

D,  $11279.75; 

F,  $1764; 

25.  $8000. 

13.  $5284.88. 

E,  $11190.75. 

H,  $1050; 

26.  $117645.    ' 

14.  $232.31. 

8.  M,  $18529.25; 

K,  $4956. 

27.  $28494.67. 

15.  $2138.05. 

N,  $6389.75. 

30.  A,  60  ft,  ; 

28.  $23544.58. 

16.  $393.42. 

9.  A,  $26666.67; 

B,  80  ft.  ; 

20.  $325. 

17.  A,  $335.34; 

B,  $27091.67; 

C,  100  ft. 

30.  $48600. 

B,  $558.90; 

C,  $2166.66. 

31.  $554.68. 

31.  $960. 

C,  $465.76. 

10.  A,  $5004.24; 

S2.  P,  $10229,71  ; 

32.  $34137.50. 

18.  $66500. 

B,  $2502.12. 

Q,  $10245.54. 

Art.  639.] 


ANSWERS. 


359 


S3.  A,  $3847.56; 

10.  $636.58; 

17.  £10.  Os.  9d. 

Cr.  B,  $12695 

B,  $4902.44. 

$637.45. 

18.  $102. 

Cr.  C,  $12695 

34.  A,  $12346.82; 

19.  $433.93. 

Cr.  Sundry 

B,  $58.77.67. 

Art.  662. 

20.  $889.36. 

Creditors, 

35.  A,  $2505.74; 
B,  $17307.58; 
C,  $16723.84. 
36.  A,  $17527.74; 
B,  $20323.76; 
C,  $6310.43  ; 
Will  lose  $140 
37.  A,  $7331.20; 
B,  $4950.50. 
38.  A,  $4932.38; 
B,  $3497.25; 
C,  $6570.37. 

1.  $131.90. 

2.  $137.20. 
3.  $116.36. 
4.  $1638.40. 
5.  $0.054; 

$0.072; 
$0.104. 
6.  $0.655. 
7.  $356. 

8.  $13472. 
9.  $10469.12. 
10.  $3700; 

21.  $261.83. 
22.  120  bbl. 
23.  135458  Ib. 
24.  $676.77. 
25.  0. 
26.  Loss,  $500. 
27.  30}-f%. 
28.  $500. 
29.  $35.58. 
30.  $137.26. 
31.  £3  19s..  lOd. 
32.  Sept.  19; 
$4950. 

$4050. 

Art.    667. 

1.  36.      5.  81. 
2.  48.      6.  126. 
3.  55.      7.  125. 
4.  72.      S.  144. 

Art.  671. 

1.  29.      5.  47. 
2.  37.      6.  6. 
3.  31.      7.  23. 
4.  43.      8.  1. 

Art.  645. 

1.  72^; 
*       $5760. 
2.  $702000; 

11.  $9080;' 
$29080. 
12.  $996.80. 
13.  $717.68. 

33.  $5078.72. 
34.  $851.96. 
35.  $925.62; 
$1074,38. 

Art.  674. 

1.  $9715.20. 

2.  $5187.21. 

$35100. 
S.  $35625. 

4.  $77400; 
$64000; 

14.  $12379.12. 

15.  $342.67; 
$26.98; 

$296.57. 

'  $1087.17. 
37.  $72.22; 
$73.89; 
^72  88 

S.  $13648. 
4.  $5631.50. 
5.  $4051.44. 

6.  $1474.27. 

$86000. 
5.  $21375; 
$475000. 

16.  $42919; 
$29169.33. 
17.  $4190.03; 

38.  $4166.30; 
$4160.47. 

7.  $3426.67. 
Art.  677. 

6.  $3.338,100. 
7.  26.26^  : 
$23,944,096.50 
$1,213,591.50. 

$3084.60; 
$933.35; 
54  yr. 

18.  $152.79; 

'  $5181.19.' 
40.  $4165.03. 
41.  $630.97. 

42.  $2474.07. 

2.  $261.58. 
4.  $1327.61. 
S.  $549.51. 
9.  $3260.51. 

8.  $1,433,831.70. 
9.  $1047.54;' 
$427.84. 
10.  $2313.65. 
11.  12^; 

$77.86. 
19.  $13.63. 
20.  $2655.65. 

Art.  663. 

43'.  $9825.03. 
44.  17651.29  fr. 

45.  $1889.22. 
46.  $7.18;  $2.39. 
47.  $114.75. 

Art.  682. 

2.  $120. 
3.  $72. 
4.  $6.20. 

c      (£917 

C.'r/ 

1.  104J-. 

48.  Aug.  29; 

O.    «p/*f. 

A    64      /    ' 

2.  21659.6. 

$5359.35. 

Art.  687. 

±4.    4j|jg-/0. 

S.  40320. 

49.  Jan.  14,1883; 

Art.   650. 

4.  $164.52. 
5.  16°  48'  15". 

$1409.40. 

50.  $1409.37. 

1.  $50.17. 
2.  $36.60. 

1.  $374.60; 

6.  16666|  sq.  yd. 

51.  $320.31. 

3.  $lo9. 

$374.93. 

7.  £69  Os.  6d. 

52.  $1625. 

irt    60O 

2.  $669.35; 

8.  $140.25. 

53.  $425. 

ff)       KQs* 

$671.70. 
S.  $25.14. 

9.  381-sq.yd.; 

$24.15. 

54.  $37000. 

55.  $81.25. 

Z.    OoC. 

3.  35c. 

/       PtO^ 

4.  $755.28; 

10.  £19  3s.  6d. 

56.  $4224. 

^.    OoC. 

$757.38. 

11.  $5567.50; 

57.  $128.21. 

^r«.  603. 

5.  $462.10; 

$4.443,  wood; 

58.  $295.75. 

$464.38. 

$.486,  grain. 

59.  $208.50. 

2.  2,1,5,1; 

6.  $190.91; 

12.  $1803.07. 

60.  B,  $5011.83; 

3,  2,  3,  5. 

$191.04. 

13.  $.8392. 

C,   $1794.53; 

3.  3,  4,  5,  1  ; 

7.  $96.15. 

14.  £219375; 

Dr.  Mdse.,etc., 

4,  3,  1,  5. 

8.  $328.60; 

$23.72; 

$12410; 

4-  1,5,6,1; 

$329.37. 

$1067588.44. 

Dr.   Sundry 

5,  1,  1,  6. 

9.  $557.31; 

15.  $45.73. 

Debtors,    > 

5.  1,  1,  3,  6,  1; 

$557.86. 

16.  $430.72. 

$17030; 

3,  3,  1,  1,  6. 

360                                                       AXSWEKS.                                          [Art.  695. 

Art.  695. 

70.  716. 

75.     5.     5.     S3. 

18.  502.656  A. 

2.  10,  10,  45. 

IS.  15.25. 

77.  35.03  in. 

3.  30,  90. 

4.  12,48,  48; 
36,  36,  108. 
5.  15  15. 

7^.  1994. 
75.  20.78. 
16.  2732. 
77.  19683. 

JLr«.  724. 

1.  18.9875  A. 

Art.  730. 

0.  6.8027^; 
$493.20. 

6.  9,15,3; 
16,  4,  20. 

Art.  697. 

2.  5,  60,  15,  30; 

£:&&,* 

00.  208.71  ft. 
07.  1.41421. 

-4»*.  705. 

0.  1.625  A. 
3.  290.47  sq.  rd. 
4.  34  A.   145  sq. 
rd.  25  sq.  yd. 
8  scj.  ft.  108 
sq.  in. 

3.  $225. 
4.  Vr.  $3581.16; 
C  p.  $3681.16; 
Adj.  rec.  $100. 
5.  $4869.57. 
6.  Ship  receives 

8,  52,  24,  26. 

0.  18. 

5.  1680.6ft. 

$1888.08; 

4.  33,  12,  5. 

3.  2.2. 

6.  471.24ft. 

Ar.  $964.23- 

Art.  701. 

4.  68. 

5.  78. 

7.  5541.78  sq.  ft. 
8.  14.7  sq.  ft. 

Bp.  $1472.82; 
C  p.  $496.94; 

2.  33. 

6.  82. 

9.  3525  gal. 

D  p.  $1032.55; 

3.  121. 

7.  196. 

70.  5.32  cu.  ft. 

Adj.  rec.  $150. 

4.  135. 

8.  2.13 

77.  15053£  tons. 

7.  C  p.  $1200.79; 

5.  216. 

9.  23.9. 

70.  59.9  in. 

S  rec.  $974.60: 

6.  218. 

70.  462. 

13.  466.69  ft. 

Adj.  rec.  $150; 

7.  255. 

77.  6.54. 

14.  75.3984  cu.  ft. 

Agent  receives 

8.  294. 

70.  75.8. 

75.  6  ft.  6  in. 

$76.19. 

9.  312. 

13.  878. 

16.  32  yd. 

8.  G.A.,  $516.81; 

70.  345. 

14.  82. 

77.  45.14rd.; 

S,  $638.79; 

11.  37.5. 

75.  47  ill. 

141.8  rd.                   O,  $789.09. 

TESTIMONIALS. 


HENRY  C.  and  SARA  A.  SPENCER,  Spencerian  Business  College, 
Washington,  D.  C.—lt  is  an  admirable  text-book  ;  analytical,  logical,  practical,  and  accu- 
rate. We  use  it. 

CURTISS  &  CHAPMAN,  Curtiss  Business  College,  Minneapolis.—  We  have 
been  using  your  book  from  the  first ;  and  the  more  we  use  it,  the  better  we  like  it.  It  is  the 
best  Arithmetic  of  which  we  have  knowledge. 

JOHN  R.  CARNELL,  Prin.  Alb.  Bus.  College,  Albany,  N.  Y.— We  find  the 
Packard  Arithmetic  in  every  way  adapted  to  our  requirements. 

GEO.  W.  SPENCER,  Prin.  Bus.  College,  Providence,  R.  T.— We  have  used  the 
Packard  Arithmetic  from  the  first  issue,  and  have  obtained  better  results  than  with  any  other 
book. 

W.  I>.  MOSSER,  Prin.  Bus.  Col.,  Lancaster,  Pa.— We  have  used  it  from  the 
first,  and  have  found  it  satisfactory  in  every  detail. 

W.  T.  WATSON,  Prin.  Bus.  Cot.,  Memphis,  Tenn.— We  have  used  it  since  its 
first  appearance,  and  could  not  suggest  an  improvement. 

T.  f7.  CA\TON,  Prin.  Cotn.  College,  Minneapolis,  Minn.— We  are  fully  con- 
vinced that  it  is  the  best  work  in  print  for  Commercial  Colleges. 

M.  MacCORMICK,  Prin.  Bus.  Col.,  Guelph,  Ont.—l  have  no  hesitation  in 
pronouncing  it  the  best  Commercial  Arithmetic  that  has  come  under  my  notice. 

J.  A.  &  M.  H.  HOLT,  Oak  Ridge,  N.  C.—We  are  using  the  Packard  Arithmetic 
with  great  and  entire  satisfaction. 

C.  T.  MILLER,  Prin.  Bus.  Col.,  Newark,  N.  «7.— We  have  used  your  series  of 
Arithmetics  since  their  first  publication.  The  results  have  been  satisfactory.  They  cover 
everything  desired  in  business  calculations. 

G.  W.  BROWN,  Business  Colleges,  Jacksonville,  Peoria,  Decatur,  and 
Grtlesburg,  III.— The  Packard  Arithmetic  is  used  in  all  my  colleges  with  very  satisfactory 
results. 

0.  P.  DE  LAND,  Prop.  Bus.  Col.,  Appleton,  Wis.—It  is  the  best  Commercial 
Arithmetic  I  have  ever  used. 

W.  K.  MULLIKEN,  Prop.  Bus.  Col.,  St.  Paul,  Minn.— I  have  used  it  since  its 
first  edition,  and  consider  it  unequaled.  It  is,  without  doubt,  the  best  book  for  high  schools 
and  colleges  that  has  yet  appeared. 

1.  E.  SAWYER,  Luther,  Mich.— It  is  the  best  Arithmetic  in  the  market.    It  is 
devoid  of  all  puzzling  questions  and  problems,  has  the  shortest  and  most  comprehensive 
methods,  and  is  simple  and  direct. 

B.  F.  MOORE,  Bus.  Col.,  Atlanta,  Ga.—  I  regard  it  the  best  work  published,  both 
for  the  class  room  and  the  counting-room. 

A.  E.  MACKEY,  Bus.  Col.,  Genera,  N.  Y.—  It  has  been  our  favorite  since  its  first 
publication,  and  we  have  no  desire  to  change.  Aside  from  its  merits  as  a  class-book,  it  con- 
tains a  vast  amount  of  information  for  the  accountant  and  business  man. 


11  TESTIMONIALS. 

HfcKAY  &  FARNEY  Bus.  Col.,  Winnipeg,  Man.— Have  used  it  since  ils  first 
publication.  It  is  just  suited  to  business  college  work. 

ALBERT  C.    BLAISDELL,  Prin.   Commercial    College,   Loivell,  Mass.— I 

have  used  many  kinds  of  Arithmetics  in  the  last  ten  years,  but  all  were  found  wanting  except 
Packard's.    This  just  fills  the  bill.    In  my  judgment  it  is  without  an  equal. 

S.  S.  GRESSLY,  Prin.  Business  College,  McKeesport,  Pa.—  We  have  used  the 
Packard  Arithmetic  since  its  first  appearance,  and  believe  it  to  be  decidedly  the  best  book 
published.  The  revised  edition  cannot  be  too  highly  commended. 

C.  IF.  ROBBINS,  Central  Business  College,  Serial ia,  Mo.—  We  have  used  your 
book  since  its  revision,  and  think  it  the  best  Commercial  Arithmetic  published. 

CHAS.  FRENCH,  Prin.  Business  College,  Boston,  Mass.— The  longer  we  use 
it,  the  better  we  like  it. 

GEO.  S.  BEAN,  Prin.  Business  College,  Peterborough,  Ontario.— We  use 
your  book  altogether,  and  find  it  very  complete.  I  consider  it  the  best  work  I  have  seen. 

E.  C.  A.  BECKER,  Worcester,  Mass.— For  the  last  ten  years  I  have  been  on  the 
alert  for  a  book  that  would  give  the  best  results,  and  can  find  nothing  better  than  the  "  Pack- 
ard." It  is  all  business— simple,  complete,  and  free  from  mathematical  puzzles. 

W.  N.  FERRIS,  President  Industrial  School,  Big  Rapids,  Mich.-A.fev/ 
years  ago  I  examined  every  Commercial  Arithmetic  that  had  been  published  in  America  ;  as  a 
result,  adopted  the  "Packard."  I  do  not  hesitate  to  say  it  leads  them  all. 

MESSRS.  EATON  &   BURNETT,  Business   College,  Baltimore,  Md.—We 

use  your  book  because  we  think  it  the  best  book  in  the  market. 

«7.  T.  MURFEE,  Prin.  Military  Institute,  Marion,  Ala.— We  use  your  Arith- 
metic, and  regard  it  superior  to  any  we  have  ever  seen. 

E.  M.  HUNTSINGER,  President  Business  College,  Hartford,  Conn.— I  have 
used  the  Packard  Arithmetic  since  its  first  issue.  It  meets  the  demands  of  the  aspiring  busi- 
ness mathematician  as  does  no  other  work.  Our  pupils  are  delighted  with  it. 

J.  T.  JOHNSON,  President  Business  College,  Knoxville,  Tenn.— I  have  used 
the  Packard  Arithmetic  since  its  first  publication,  and  like  it  better  than  any  other  Arithmetic 
I  have  ever  examined.  It  is  plain,  practical,  and  concise. 

WILLIAMS   &    BARNES,     Commercial    College,    Iowa    City,    Iou'a.—We 

adopted  your  Arithmetic  several  years  ago,  because  we  considered  it  the  best  book  published 
for  commercial  colleges.    We  expect  to  continue  its  use. 

E.   L.   McILRAVY,  Pres.  Business   Unirersit?/,  Kansas  City,  Mo We  use 

the  New  Packard  Arithmetic  because  we  consider  it  the  best  work  of  its  kind. 

JESSE  SUMMERS,  Pres.  Normal  College,  Abingdon,  III.— We  are  using  your 
book,  and  are  well  pleased  with  it. 

T.  D.  GRAHAM,  Business  College,  Nashville,  Tenn.— I  have  used  your  New 
Commercial  Arithmetic  since  it  was  first  issued,  and  can  speak  authoritatively  of  its  merits 
It  is  a  most  excellent  book. 

T.  R.  BROWNE,  Prin.  Business  College,  Brooklyn,  N.  Y.—It  is  not  simply  a 
little  better  than  any  other  book,  but  beyond  comparison  with  other  books.  I  have  used  it 
from  the  beginning. 

C.  A.  FLEMING,  Prin.  Northern  Business  College,  Owen  Sound,  Ont.—We 

have  used  your  Arithmetic  since  its  first  publication,  and  have  found  it  in  all  respects  com- 
plete.   It  is  just  the  book  for  the  business  college  student. 

E.  «7.  GANTZ,  President  Normal  College,  Humestown,  Iou'a.—l  have  used 
your  Arithmetic  since  its  first  appearance,  in  the  Commercial  Department  of  this  institution, 
and  have  seen  none  that  I  would  exchange  it  for.  I  consider  it  the  best  book  in  use. 


TESTIMONIALS.  Ill 

A.  J.  WARNER,  President  Business  College,  Eltnira,  N.  Y.—l  regard  your 
Arithmetic  the  best  in  the  market.  You  have  made  a  great  hit  in  its  preparation. 

THOS.  J.  STEWART,  Prin.  Stewart  &  Hammond  Business  College,  Tren- 
ton, N.  J.—l  regard  the  New  Packard  Commercial  Arithmetic  as  far  superior  to  any  similar 
work  now  published.  The  longer  we  use  it,  the  better  we  like  it. 

G.    M.    NEALE,  Pres.   Commercial    College,  Fort    Smith,  Ark.— We  use  the 

Packard  Arithmetic  in  our  school  because  we  think  it  the  best  book  to  be  had. 

C.  W.  BUTLER,  Supt.  I'nblic  Schools,  Defiance,  Ohio.— Your  Commercial 
Arithmetic  has  been  used  in  our  High  School  since  its  first  publication.  It  is  a  practical, 
common  sense  book.  I  know  of  nothing  better. 

A.    H.    HINMAN,    Prin.    Business    College,    Worcester,    Mass.— I   use  the 

Packard  Commercial  Arithmetic,  like  it,  and  heartily  commend  it,  because  its  work  is  so  com- 
prehensive and  practical,  and  so  admirably  prepared  for  use  in  business  schools. 

SHANNON  &  BISSON,  Proprietors  Business  College,  Muskegon,  Mich.— 

We  are  using  your  Arithmetic,  and  think  it  the  best  book  to  be  found  for  universities,  business 
colleges,  and  high  schools.  It  is  thoroughly  practical  and  comprehensive. 

IT.  D.  McANENET,  Pt-in.  Business  Department,  Drake  University,  T>fs 
Moines,  la.— We  have  used  the  Packard  Arithmetic  since  it  was  first  published.  For  the  gen- 
eral purposes  of  business  schools  it  surpasses  any  other  work  of  the  kind  that  I  have  seen. 

TEMPLE  &  HAMILTON,  Business  College,  San  Antonio,  Texas.— We  have 
adopted  your  book  as  the  most  thorough  and  teachable  text-book  in  the  market.  It  is  a  work 
of  superior  merit. 

J.  M.  ME  HAN,  Prin.  Business  College,  Des  Moines,  Tow  ft. —The  New  Pack- 
aid  Arithmetic  has  been  in  use  in  my  school  since  its  first  publication,  and  it  fully  meets  all 
wants.  It  is  admirably  graded,  practical  in  all  ways,  and  fully  up  to  the  times. 

«7.  R.  GOODIER,  Prin.  Business  College,  Port  Huron,  Mich I  have  used 

the  New  Packard  Arithmetic  since  it  was  first  issued,  with  the  very  best  results.  From  the 
first  page  to  the  last  the  good  of  the  student  is  kept  constantly  in  view. 

BENNETT  &  GREEK,  Proprietors  Morrell  Tnstittite,  JTohnstown,  Penn.— 

We  have  used  your  book  in  our  school  for  over  three  years,  and  know  of  no  other  that  so  fully 
meets  modern  requirements.  We  especially  commend  its  short  and  easy  methods  and  its 
practical  examples. 

RICKARD  &  GRTJMAN,  Minneapolis  School  of  Business,  Minneapolis, 
Minn. — The  Packard  Arithmetic  cuts  a  prominent  figure  in  all  our  calculations.  It  teaches 
everything  from  round  numbers  to  square  root.  No  student  can  go  through  it  and  come  out  a 
cipher.  It  adds  to  his  capacity,  multiplies  his  faculties,  and  divides  his  merits.  In  its  whole 
number  of  pages,  we  find  no  fraction  of  waste,  and  its  decimals  are  always  to  the  point.  Our 
interest  in  this  book  is  compounded  every  year. 

ROHRBOUGH    BROS.,  Proprietors  Commercial  College,  Omaha,  Neb 

We  have  used  your  Arithmetic  since  its  first  publication.  Did  we  not  believe  it  to  be  the  first 
in  the  market  we  should  not  use  it.  It  should  be  introduced  into  every  first-class  commercial 
college  in  the  country. 

E.  H.  FRITCH,  Prin.  Business  College,  Wichita,  Kansas.— We  are  using 
the  New  Packard  Commercial  Arithmetic,  and  believe  it  to  be  the  best  work  of  its  kind  ever 
published. 

VARNUM  &  BENTON,  Proprietors  Business  College,  Denver,  Colo.— We 
have  used  your  book  for  the  last  four  years,  and  consider  it  in  all  respects  the  best  book  before 
the  public. 

J".  W.  HALEY,  Fort  Edtvard,  N.  Y. — Your  New  Arithmetic  is  in  every  respect 
complete,  and  just  what  is  needed  in  every  business  college. 


IV  TESTIMONIALS. 

E.  L.  ELLIOTT,  Waterloo,  Iowa.— The  New  Packard  Commercial  Arithmetic  is  a 
jewel. 

A.  J.  RIDER,  Prin.  Easiness  College,  Trenton,  N.  «7.— It  is  decidedly  the 
best  work  extant. 

W.  II.  ROGERS,  Cashier  Nassau  Sank,  New  York.— it  is  the  best  book  on 
the  subject  that  I  have  ever  seen. 

O.  G.  NEUMANN,  Prin.  Business  College,  Austin,  Texas.— We  have  seen 
and  examined  many  other  books,  but  this  takes  the  lead. 

T.  J.  rRICKETT,  Prin.   College  of  Commerce,  Philadelphia,  Penn.—The 

use  of  your  book  for  the  past  six  years  has  only  strengthened  my  opinion  that  it  is  the  best 
work  of  its  class.  My  teachers,  without  exception,  are  enthusiastic  in  their  praise  of  the 
"PACKARD." 

HO  WELL  B.  PARKER,  Prin.  Academy,  Hampton,  Ga.—  The  New  Packard 
Commercial  Arithmetic  has  been  severely  tested  in  my  school,  and  grows  better  every  day. 
Having  made  a  specialty  of  Arithmetic  for  twenty  years,  I  can  say  that  yours  is  the  best  that  1 
have  ever  used. 

T.  C.  STRICKLAND,  Prin.  Acadwny,  East  Greenwich,  R.  I.— It  is  admi- 
rable throughout,  and  worthy  of  special  attention  for  its  treatment  of  interest,  equation  of 
accounts,  and  partnership  settlements. 

G.  R.  RATHBUN,  Business  College,  Omaha,  Neb.— It  is  practical  and  fully 
adapted  to  business  college  work.  No  book  could  take  its  place. 

-8.  A.  DRAKE,  Clark's  Btisiness  College,  Erie,  Penn.—It  fully  meets  the 
requirements  of  all  grades  of  commercial  classes. 

*7.  E.  GUSTITS,  Prin.  Business  College,  Rock  Island,  III.— It  is  the  most  com- 
prehensive and  tnorough  treatise  upon  the  subject  of  commercial  calculations  ever  published. 

E.  A.  HALL,  Prin.  Business  College,  Logansjtort,  Ind. — I  have  taught  com- 
mercial arithmetic  for  over  twenty-five  years,  and  consider  the  New  Packard  Commercial 
Arithmetic  superior  to  any  other  book  I  have  used.  It  is  modern,  progressive,  and  full  of 
common-sense  problems  such  as  are  used  in  every-day  business  affairs. 

THOS.  H.  SHIELDS,  Prin.  Business  College,  Troi/,  N.  T.— We  have  used 
your  Arithmetic  in  our  school  since  it  was  first  published,  and  consider  it  the  best  in  the 
market. 

E.  G.  G  UION,  Prin.  Business  College,  Washington,  Penn.—We  use  your  book, 
and  consider  it  a  most  valuable  text-book.  The  absence  of  catch  questions  has  left  room  for 
the  practical  and  useful,  which  are  well  supplied. 

H.  I.  MATHEWSON,  Milford,  Conn.— The  New  Packard  Commercial  Arithmetic 
fully  meets  my  best  expectations. 

W.  F.  L.  SANDERS,  Supt.  Schools,  Connersville,  Ind.— We  know  of  no  better 
Arithmetic  than  the  New  Packard  Commercial.  The  school  that  adopts  it  will  keep  it. 

MISS  E.  A.  TIBBETTS,  Business  College,  Salem,  Mass.— I  have  found  it  a 
most  comprehensive  and  satisfactory  book  from  beginning  to  end. 

L.  R.  WALDEN,  Prin.  Business  College,  Austin,  Texas.— I  know  of  no  work 
that  so  nearly  meets  with  my  idea  of  a  text-book  on  this  subject. 

C.  BAYLESS,  Prin.  Business  College,  Dubuque,  Iowa.— We  have  used  it  from 
the  beginning.  It  is  a  work  of  superior  merit. 

,7.  A.  McMAHON,  Prin,  Business  College,  Beaver  Falls,  Pa.— In  my  opinion 
it  is  the  best  Arithmetic  of  the  kind  that  is  published. 

L.  A.  GRAY,  Prin.  Business  College,  Portland,  Me.—  The  New  Packard  Com- 
mercial Arithmetic  has  been  used  in  this  college  since  it  was  first  published.  It  gives  entire 
satisfaction,  and  seems  to  increase  in  favor  the  longer  we  use  it. 


YC  2244! 


M98496 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


